(written from a Production point of view)
Digitally remastered episodes are re-releases with audio and video digitally, or computer-aided, enhanced. For video this currently means enhancing to either 720p (usually for television productions), or 1080p (usually for motion picture productions) high-definition (HD) standards, typically for Blu-ray Disc releases. The 720p and 1080p refer to the horizontal resolution lines on a viewing screen. Anything upward from 720p is considered HD as opposed to the old standard-definition (SD) of 480i (for America) and 576i (for Europe). Even higher HD resolution formats are currently under development. Digital audio enhancement usually involves upgrades from any older sound system or format to DTS standards, which could be either Dolby TrueHD or DTS-HD Master Audio.
Early Star Trek: The Original Series remastering
While the current understanding of the Star Trek remastering projects concerns the projects CBS Television Studios/Paramount Pictures embarked upon in the 2000s, essentially starting with the 2001 release of Star Trek: The Motion Picture (The Director's Edition), Paramount Television had already embarked upon an early Star Trek: The Original Series remastering project in the early 1990s. Called "digitally mastered" (as indicated on the below-mentioned UK VHS box-art) at the time, new, cleaned-up and color corrected, transfers were produced from the old masters. The operation was especially commissioned for the 1992-1993 Japanese Original Series LaserDisc collection releases, Star Trek - Log 1, Log 2 and Log 3. Picture quality wise, it was the best edition commercially available until the advent of the 2007 TOS-R Season 1 HD DVD version, the earlier DVD versions still making use of the original masters. These Japanese transfers were used twice more, for the lesser known UK 1996-1998 VHS re-release, and the 2007-2008 Star Trek: The Original Series - The Collector's Edition DVD/Magazine combo release.
Little information is available on this project, as this version was never released on any home media format in America or mainland Europe. In 2012 it was revealed that stills originating from this particular project were the ones that were included as illustrative backdrops in the 2010 reference book, Star Trek: The Original Series 365. The authors have elaborated, "Most of those wonderful TOS photos that had never been seen before came from a very expensive project that the licensing department conducted many years ago. In order to find new images for the licensees, the department struck new 16-millimeter copies of the TOS masters—which were then carefully cut up, frame by frame, and mounted into slides." 
Star Trek: The Motion Picture - The Director's Edition
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In 2001, the aforementioned Director's Edition of Star Trek: The Motion Picture was released on DVD. While the production did not, strictly speaking, confirm to the current understanding of remastering, i.e. the upgrade from SD to HD, it was in essence a remastered production. It has received a new, upgraded soundtrack. (Star Trek: The Magazine Volume 2, Issue 8, p. 62) Most notably, however, was that many scenes were digitally enhanced and visually changed with the addition of computer-generated imagery (CGI). Striking examples of these were the enhanced air tram station as well as the first and only full view of V'Ger, which was not there in the original version. The CGI for this production was provided by effects house Foundation Imaging. (Star Trek: The Magazine Volume 2, Issue 8, pp. 52-61)
Star Trek: The Original Series
On 31 August 2006, CBS Paramount Television announced that, in celebration of the 40th anniversary of Star Trek: The Original Series would return to broadcast syndication for the first time in sixteen years. Beginning with "Balance of Terror", each of the series' 79 episodes were digitally remastered to 1080p HD video, and DTS-HD Master Audio 7.1 Surround audio standards , with all newly (re-)created visual effects and music. The refurbished episodes were converted from the original film to a high-definition format similar to that used on Star Trek: Enterprise. The opening theme was re-recorded in digital stereo with new vocals by Elin Carlson, and William Shatner's opening monologue was remastered from the original elements. Most notably, though, many of the visual effects were recreated using CGI by CBS Digital. The opening credits sequence was revamped, several matte paintings received a CGI face-lift, and spaceship exteriors including the Romulan Bird-of-Prey and the Klingon battle cruiser were recreated using state-of-the-art digital effects. However, for some reason, the credits at the end of the episode were not remastered and digitally enhanced. The new CGI Enterprise was based on the exact measurements, originally taken by Gary Kerr, of the original model, which is on display in the gift shop of the Smithsonian Institution's National Air and Space Museum in Washington, DC.
In an interview with TVGuide.com, project supervisor Michael Okuda said, "We're taking great pains to respect the integrity and style of the original... Our goal is to always ask ourselves: What would Roddenberry have done with today's technology?" Denise Okuda and Dave Rossi were also involved with this relaunch of the original series; Niel Wray is the visual effects supervisor. A Q & A with the production staff was posted by the official Star Trek website on 6 September 2006. A video preview and interviews with people involved in the project can be viewed by clicking the link to the left of the page.
The first few episodes were rushed, as CBS only gave its team "one month to deliver the first two episodes with over 120 new effects shots." Starting with "The Trouble with Tribbles", a new, improved Enterprise model was used. Members of the effects team have commented that they may go back to the earlier episodes and re-render the ship scenes with the new model  , though that did not come to fruition.
When TOS cast member Leonard Nimoy heard about these changes in special effects, Nimoy simply responded "Shame on them" for changing the effects, saying that it was "out-of-bounds" for them to do that. However, after viewing a remastered episode, reportedly he was quoted saying, "I'm amazed." 
The seasons were to be released in an HD DVD/DVD combination set, with season one released in November 2007. The release of seasons two and three were canceled in February 2008 due to the decline of HD DVDs in comparison to rival format Blu-ray. However, the second season was released on regular DVD on 5 August 2008.  all three seasons are available on Blu-ray Disc, as well as through Apple's iTunes Store (along with a "best of" collection for budget-minded fans), with select episodes available on Microsoft's Xbox 360 Video Marketplace. Unlike the HD DVD releases, the Blu-ray editions also include the original versions of each episode, allowing the original special effects and images to be seen in high-definition for the first time.
CBS Digital wrapped up their visual effects work on TOS Remastered just before midnight on 21 April 2008. The last episode they worked on was the show's first pilot, "The Cage". The last shot rendered was of the Enterprise "sailing off into the unknown at the end of the episode." 
Star Trek films
The success of the remastered Original Series, as well as the upcoming re-launch of the Star Trek franchise in 2009, encouraged Paramount Pictures to push forward with the remastering to 1080p HD video, and Dolby TrueHD 7.1 Surround audio standards  of their movie properties. The remastered versions of the first ten Star Trek films were released from 2009 to 2010 in both the DVD and Blu-Ray disc formats. According to the box-art of the remastered DVD version of Star Trek II: The Wrath of Khan, it required the most extensive restoration work, and was "fully restored in high definition with brilliant picture quality", whereas the remaining nine were just "digitally remastered", due to the fact that the original master print "was in terrible shape", according to Director Nicholas Meyer (Audio commentary, Star Trek II: The Wrath of Khan (2009 DVD))
A striking detail was, that when it came to the remastering of The Motion Picture, being one of the last released in 2010, only the original theatrical version could be remastered. The reason for this was that Foundation Imaging's additional CGI elements for the Director's Edition were no longer available for upgrade (in this case re-rendering to a higher resolution), due to Paramount's failure to maintain ownership of these elements. Reconstructing the CGI effects from scratch was cost-prohibitive. Nevertheless, according to one reviewer, the sound quality was superior to all the others. What this reviewer had overlooked, however, was, that the soundtrack had already received an extensive overhaul in 2001 for the "Director's Edition", a privilege the other nine movies did not enjoy. On the other hand, the eleventh, 2009 movie Star Trek did not need any remastering, as it was already shot in HD standards.
It is unclear which company has been responsible for the remastering of the movies.
Star Trek: The Next Generation
After several months of speculation and partial confirmation, on 28 September 2011 (the 24th anniversary of the series premiere), StarTrek.com announced that The Next Generation would be remastered to, again 1080p HD video, and DTS-HD Master Audio 7.1 Surround audio standards, for release on Blu-ray Disc and eventual syndication, starting in 2012.
The remastering process requires returning to the original film negatives, as all editing and visual effects compositing took place on video, which lacks the required resolution for high-definition. This collection of footage, totaling over 25,000 reels, is being re-edited exactly as was done for original airing. Visual effects are being entirely recompositioned, and not upconverted from the original videotape. Audio is the form of 7.1 DTS Master Audio.
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As with the remastering of The Original Series, Denise and Michael Okuda are serving as consultants. Unlike its predecessor, none of the visual effects were slated for replacement with newly conceived CGI, or as Project Consultant Mike Okuda has put it, "We love the approach that CBS took for this project. We take the original film elements and put them together in a new way. The material still has all the details and they are beautiful. And the new visual effects are really the old visual effects but a lot more beautiful than you have ever seen them."  However, some use of CGI was made. This was restricted to correcting continuity details, or for replacing original elements that were either lost, such as in the case of the Crystalline Entity, or technically impossible to upgrade. The most notable examples of the latter case were the replacements of long shots of planets, which were originally executed as matte paintings, by newly created CGI effects, though Digital Artist Max Gabl took care they approximated their original appearance. Gabl stated, "I think most of them are total recreations. Because the planets we’re looking at from the original [TNG] series are very low-res and blurry. There’s no way to put more detail into those, so it’s basically all recreation. Mike Okuda tells us exactly what we need in there, and it’s just back and forth – playing it and seeing what the details are going to look like and then I put them in, compare with the old, [Mike will] look at it, I’ll make the changes and that’s how it goes." 
CBS Digital was again appointed lead company in the remastering project, as well as being made responsible for the creation of the replacement CGI effects.  In order to get the most exposure for the project, CBS Studios reached out to the Star Trek fan community and gave access to the project to the administrators of the TrekCore.com-blog for them to keep the community appraised of every aspect of the project, among others by publishing sneak previews and the like. TrekCore's Adam Walker was given access to production staffers and actors alike for conducting in-depth interviews that were published on the blog. TrekCore is cooperating with another blog, Ex Astris Scientia, where MA admin Jörg broke down each episode for the differences and occasionally received input from production staffers to explain detail issues. 
The first release for the high-definition Next Generation was a sampler disc, Star Trek: The Next Generation - The Next Level, released on 31 January 2012,  and the full series followed. The TNG Season 1 Blu-ray was released on 24 July 2012 and the TNG Season 2 Blu-ray was released on 4 December 2012. The TNG Season 3 Blu-ray is scheduled to be released on 29 April 2013.
Highly anticipated, the first season release was well received and did very well in sales, selling 95,435 copies in the first week after release in the US alone.  By then, CBS had already decided to speed up the release schedule for the remastered series. For CBS to be able to do so, they had to sub-contract, considering the amount of work the remastering entailed, other companies to do the work on a season alternating basis. Independent company HTV-Illuminate was awarded the commission for the upgrade of the second season. However, their work was less well received, as many considered their efforts sub-par to that of CBS Digital.  Though not having confirmed if this was in any way related, CBS has made the decision to go with another effects house, Modern Video, for the remastering of season four, while simultaneously keeping a tighter rein over the work. 
Documentaries and specials
- TOS Season 1 Blu-ray–special feature, "Spacelift: Transporting Trek Into The 21st Century"
- TNG Season 1 Blu-ray–special feature, "Energized! Taking The Next Generation to the Next Level"
- TNG Season 2 Blu-ray–special feature, "Energized! Season Two Tech Update"
- fxguidetv #161: CBS Digital & Star Trek TNG, 2012