This is the Time Machine that was built by Dimitri Allen, Bill Hawks and Claire. Its completion meant everything to Dimitri and Bill. When it was "finished", Bill proposed using Claire as the test subject. Dimitri was completely against the idea, but Bill went on without him, since he was offered a lot of money for the technology, and he needed to demonstrate the machine's viability.
The test run failed, and the time machine exploded. The blast was so powerful that it almost completely destroyed the surrounding area. The apartments next door had been demolished, and at least ten people had been killed, among them being Claire, and Clive's parents.
It was later revealed that the Time Machine had worked for a single moment before it exploded, and it had shot Claire ten years into the future (the present). But because of its incomplete nature, she had to go back to the moment of the explosion.
The time machine accident was what caused the events of Professor Layton and the Unwound Future. It should be noted that this is the only real Time Machine in the game.
The second Time Machine was a fake that was developed by Dr. Stahngun/Dimitri Allen and his assistants. It was built so Dimitri could kidnap Bill Hawks and his assistants, so he could take them to Future London. For the presentation, he had invited most of London's high society, but had also invited Professor Layton and Inspector Chelmey, so they would be lured to Future London as well. When it exploded, everyone thought that Bill and Dimitri were dead, but this was later revealed to be untrue.
The third Time Machine was another fake. Under the disguise of a giant clock in Spring and Cogg's clock shop, it was used to bring people to Future London. According to Cogg's explanation, it was able to create a wormhole that would send everyone forward or back in time.
It was later revealed that this "Time Machine" was in fact part of a giant elevator, and Future London was situated in a cavern below London. It assisted in evacuating the citizens of Future London when the Mobile Fortress became active.
The fourth and last Time Machine was the one Dimitri Allen was trying to build in Future London. The task was too large for him alone, so he kidnapped scientists from London, and told them that completing the Time Machine was their only hope of returning to their own time. Dimitri's true motives were to travel back in time to save Claire, and to rob Bill Hawks of all that he had won. This Time Machine was never seen, and never completed. It was destroyed along with Future London. When this happened, Dimitri gave up on building the Time Machine.