(written from a Production point of view)
Only mentioned as being a travel pod in scene 40 of the script of Star Trek: The Motion Picture, the design of the diminutive travel pod did had a long way to fulfillment. Though having made its first appearance in The Motion Picture, it had already been envisioned to appear in the pilot episode of the television series Star Trek: Phase II, "In Thy Image". Nevertheless, when that project was upgraded to a full-blown movie project, a redesign was called for. In the end it resulted in the fact that the design was one of the designs debuting in a Star Trek live-action production that, until the re-invented Star Trek film of 2009, introduced the most new space faring designs at once, nine in this case.
The latest script requirements of "In Thy Image", the pilot episode of the television project Star Trek: Phase II, did provide a inspection tour by Admiral Kirk and Scotty. As such, per-visualization art was dutifully provided by Production Illustrator Mike Minor. His concepts, however, echoed the more conventional shuttle designs, as has been seen before. The early concept paintings he created for Phase II, showed a travel pod numbered "4". (The Art of Star Trek, pp. 152-153, 168-169)
Once Phase II was upgraded to Star Trek: The Motion Picture, the design work, including that of the travel pod, reverted to the Robert Abel and Associates Design Studio, or rather its subsidiary Astra Image Corporation. Art Director Richard Taylor took on the task of redesigning the various vehicles, but became somewhat frustrated by Gene Roddenberry's visual view dictates for the project,
"For example, he said the doors on the USS Enterprise are 12 feet in diameter and round. I said "A round twelve foot in diameter door? That's just monstrous!" then he said we want this shuttle to dock with the Enterprise . And I said "so, you are telling me I have to design a shuttle with a 12 foot in diameter door?" And he said, "Yes, I want these big grand doorways to open up onto the Enterprise". I said, "can these doors iris open?" "No, they can't iris open". I said "how can this little shuttle have a door that big and open it if doesn't iris open?" So, I finally came up with a solution that the doors when docked with a station or other spacecraft open or slide into the walls of the adjoining craft.(...)Yes, well I designed that shuttle. The first shuttles that I designed were really cool, but it just turned uglier and uglier until it had what I call that "guppy" look because Kirk had to fly around the dry-dock and "see his mistress" and you had to see Kirk and Scotty in the window while they are flying around; he had to have a big panoramic view, and it had to have a 12 foot door at the back to dock with the Enterprise. It's one ugly little spacecraft as far as I'm concerned." 
By that time Taylor had brought in Andrew Probert to help him out with the design work, "All of those models Andy Probert and myself designed. I drew the original blueprints myself and I had those blueprints before Andy came on at all. Then Andy came on and he worked on some more details and he made some suggestions and we went back and forth exchanging ideas."  Eventually, the design of the pod evolved out of another requirement of Roddenberry to have the orbital office complex designed as a cluster of many offices. "Part of his reasoning for that was to provide a cinematic surprise when one of the smaller offices broke away from the others, becoming a small space ship called, in the script, a "Travel Pod". This was the small engineer's inspection vehicle that Scotty would use to ferry Kirk over to the Enterprise for his (and our) close inspection of the new ship. My early "pod" concepts were very close to the office shapes, as required, but when Roddenberry introduced "further requirements"–i.e., made changes–it evolved to what was eventually shown on screen.", co-designer Probert explained. (Starlog photo guidebook Special Effects, vol. 5, p. 90) On his thoughts behind his design work he later elaborated:
"My original concepts had a docking hatch at the rear and a docking ring off-set (like the cockpit of the Millennium Falcon) to the right. Gene wanted the ring moved to the back as a single entry/exit with auto-docking taking the worry out of backing into a docking cavity. When I opened a discussion about the doors, he said that receiving structures would have mechanisms to pull the doors open... connecting to those three recesses in the (lighter) gray doors. Prior to that, after the pod backs in, over twenty latches rotate to an attitude perpendicular to the ring, protruding into a receiving groove. They all move forward, pulling the pod into a hard docking against its black sealing-ring. As for the single 'belly-band' light,... that was my original concept, having the entire area around the pod illuminated. Being, basically, an Engineers' inspection car, I thought a lot of light was appropriate. Douglas Trumbull wanted to break that space up and have little dividing panels on them so “they would flash, like the saucer lights in Close Encounters, as it went past camera”. As for the tiny signage, after looking at real military aircraft, I realized a lot of their surfaces were covered with various instructions, labels, and warnings. I went out and bought a bunch of models decal sheets and we all covered that model with what felt right. I knew most of them wouldn't be 'seen'... but they would be 'felt', helping to make it a bit more realistic." wbmWhile the design worked out eventually, Probert was ultimately not entirely satisfied with the craft's appearance. "I was sort of forced into that design solution," he explained. "I would have preferred something a little more 'Probertesque.'" 
Full scale mock-up
As with the previous incarnations (and subsequent ones) of small scale ships, a full scale set was constructed of the travel pod, albeit only the first forward ⅓ of it. It was used to film the live action footage of Admiral Kirk and Scotty in the travel pod as it was heading toward the Enterprise. Rick Sternbach, who at the time was a lowly uncredited production illustrator remembered an incident involving the pod,
"The story that I tell people most often is that one morning I came to work about 15 minutes late and found a note on my desk saying, "Robert Wise wants to see you immediately." I thought, "Oh no, what did I do now?" I went down to Stage 17 where they were filming the sequence when the travel pod leaves the office complex to head out to the drydock. I said, "Mr. Wise I got your note, what can I do for you?" He was very quiet, he said, "You designed the controls in this pod, right?" I said nervously, "Yes, can I help you with something on that? " He walked into the pod where Jimmy Doohan and Bill Shatner were standing around. This is like a thousand dollars a minute production time, and they were waiting for me. So, Mr. Wise stepped up to the back of the pod and he said, "Tell Scotty what to do!" Jimmy was right behind me, and I said, "OK, Jimmy, when you first walk in there's like a systems activation panel, and the lights came on. So he hit the panel, and the lights came on. Then he walked to the front of the pod and got the propulsion system started and the thrusters and the artificial gravity and whatever else, and then he pushed the button to make it go. Mr. Wise thanked me and I went back to work." (Star Trek: The Magazine Volume 2, Issue 8, p. 112)
An actual physical studio model was eventually constructed at Magicam, Inc., where it was worked upon by among others model maker Chris Ross, though Richard Taylor remained less than enamored with the eventual outcome, "Roddenberry insisted that the Enterprise have 12ft. in diameter circular doors. Yeah, that's right, massive circular doors! I asked if the doors could iris open? 'No' was the answer. So all craft that docked with the Enterprise also had to have 12ft diameter doors. Do you realize how big a 12ft circular door is? So the travel pod and the Vulcan shuttle had to have 12ft doors on them. It was a silly demand and totally impractical in every respect. It also made the Shuttle that Scottie and Kirk took on the grand tour of the Dry Dock and the New Enterprise a silly looking space shuttle. It had that giant circular door as the whole rear end of the thing." (Star Trek: Creating the Enterprise, pp. 102-103)
Apart from the master studio model, at least one very small miniature was constructed to go with the drydock model, where it was seen parked inside the structure. As far as the main model was concerned, Chief Modeler Jim Dow has stated that the scale was 2"=1' (Starlog, issue 27, p. 30).
Upon completion of the model, Andrew Probert lent a hand in the finishing, as he added the weathering onto the model. 
The travel pod of The Motion Picture was filmed on one of Douglas Trumbull's stages at Future General Corporation (FGC). (The Making of Star Trek: The Motion Picture, p. 206) Preparing the travel pod studio model for filming involved Photographic Effects Cameramen Hoyt Yeatman and Alan Harding. (The Making of Star Trek: The Motion Picture, interior color photograph section) Though the long shots of the pod were taken with footage of the model with figurines in it, there were shots in which live footage of the actors were composited into footage of the model, or as Trumbull has stated,"The live-action footage was shot in a stage against a black velvet and then rear-projected into the travel pod. The model, which was a couple feet long, had a hatch in the rear that was removable; so for the scenes where you see Kirk and Scotty inside. The rear hatch was taken out and the pod was mounted along with a rear projection unit on a rig which allowed both of them to travel down a track together. So what you're seeing is a rear-projected image through the window." (Cinefex, issue 1, p. 15)
Not all long shots of the travel pod required the technically challenging use of rear projection of the actors in the cockpit. For the long shots where discernment of the characters in the cockpit was less of an issue, in-scale figurines of Scotty and admiral Kirk were constructed, that could serve as stand-ins for the actors. (The Art of Star Trek, p. 170) Some of the exterior footage shot of the model, was reused in Star Trek II: The Wrath of Khan. Shots of the actors in the cockpit of the pod, taken for that feature, were presented full screen, so that there was no need of rear projecting newly shot footage into the model.
For its appearance in Star Trek IV: The Voyage Home, the travel pod studio model was refurbished and reused. (Trek: The Unauthorized Story of the Movies, p. 122) Model Shop Supervisor Jeff Mann stated, "We added a back half to [it]." (The Making of the Trek Films, 3rd ed., p. 68) The modifications performed at ILM entailed the removal of the backplate with the large circular airlock door, that irked Taylor so much. It was replaced with a ribbed plate with a like-wise ribbed center back-spine strut (reminiscent of the back of the orbital shuttle, that accompanied the pod in the movie) that held up a protruding bottom thruster nozzle, though, ironically, the modification was not visible in the feature. In this feature scenes of the crew in the cockpit of the pod were rear-projected in the model.
Not having been reused afterwards, the model, in its refurbished form, has escaped the 40 Years of Star Trek: The Collection and It's A Wrap! sale and auctions, and has been retained by Paramount Pictures as a tour exhibit display piece, having been on tours such as Star Trek World Tour, Star Trek: The Adventure and Star Trek The Exhibition as late as 2011.