There will be, at most, two types of time machine. The first one, and the best one. This is because anyone dedicated enough to actually build a time machine
can simply go into the future and find the best form of time travel
The first set of specs on how to build a real time machine were drafted in the 1970's by an Engineering student. They involved creating a closed loop of time (which, while tricky, is theoretically possible according to relativity) by using an infinitely long cylinder hung in space. Of course, the only purpose for building a time machine out of an infinitely long cylinder would be to go back in time and tell yourself not to waste the effort.
In a fit of logic worthy of a Douglas Adams novel, physicists then announced that it might be possible to use a cylinder that was only 1 kilometer in length, if it was made out of a superdense material. Material so dense, in fact, that the cylinder would collapse along its long axis before it could ever be completed.
And so the world continued to wait for a viable time machine, which may be the only thing really worth waiting for. It would, if you think about it, be the last thing you ever would have to wait for. This is why television has been so popular since the 1970's.
Thankfully, just such an alternative has appeared on the horizon thanks to the research of one Ronald Mallett.
Despite not having any mass, light still bends space. Mallett explained how, were light to be refracted and reflected to form a ring, it would create a spacial vortex within the circle. Then came the eureka moment: Time could be bent in the same way.
Time, being somewhat stickier than space, would require a second light beam, sent around in the opposite direction to the first. However, if the light were intense enough, time and space would swap places. This means that what an observer would sense as time would actually appear as a spacial direction to one inside the ring of effect. Walking in the correct direction, it would be possible to step out of the ring before one had actually entered it.
The titanic amounts of energy required for such a feat are daunting, but, as luck would have it, not actually that tricky. As light is slowed down, it gains inertia, and thus energy. Conveniently, Lene Hau of Harvard University has managed to slow light down to a few meters-per-second (a bit slower than the usual 300,000 kilometers-per-second light prefers to move at).
Of course, slowing light requires firing it through an ultra-cold bath of atoms called a "Bose-Einstein condensate," just a few degrees over absolute zero. Still, figuring out how to deal with temperatures that low will probably prove easier than punching wormholes through the fabric of the universe.
The current plan is to test this whole mess by constructing a ring and putting a single particle in the middle. Hopefully, as was the case in Back to the Future II, if the particle moves through time and meets itself, the universe will not end.
Further, none of this takes quantum mechanics into account. Quantum theory suggests that a time machine would magnify quantum flux to the point where it would form an intense beam of radiation, destroying whatever was passing through the ring, the ring itself, or both. However, this may not happen at all, since proving it would require a theory of quantum gravity, which would unite quantum physics with relativistic physics, which would be pretty sweet in and of itself.
The only other problem with the "temporal loop" method of time travel is that, using a given ring, it would only be possible to move back in time as far as when the ring was first turned on. This means you couldn't go back in time and hang with your favorite religious figure, kill Hitler, Beethoven or Stalin, save Jim Henson, Edward Gorey or Douglas Adams, or adopt a pet dinosaur. It also means that were we to build a fixed gate, we would immediately be bombarded by an uncountable number of time tourists, wanting to see how it all began.
New Scientist magazine for 19 May 2001 (ironically, the day after I made this writeup).
Continuum: Role Playing in the Yet (Yes, it's a game, but they researched time travel theory well).
The Back to the Future trilogy from Amblin Entertainment and MCA/Universal