Wow! Glad to see some discussion on this topic so suddenly!
Jack Keiser:
See, the Grandfather Paradox demonstrates that there is a contradiction that arises when you are allowed to travel into your own past and have Free Will. Thus by removing either one of these conditions, we remove the contradiction. So there are two possibilities:
1) Time travel into the past is impossible.
or
2) You can travel into the past, but do not have Free Will while there.
The second option may seem a bit odd, but then again who's to say you have Free Will right now? See my Free Will thread for a completely separate discussion on that topic.
The way I like to best explain things is like the movie Terminator 3. I don't feel like using spoiler tags, so if you haven't seen the movie, just stop reading now.
The machine travels back in time to the present from the future to try and change things. Long story short: despite their efforts to try to change the future, their actions directly caused the future to turn out exactly the way it did. They all thought that they were going to change the future, but ended up just playing out exactly what already happened and never changed anything. Because changing the past would be contradictory. (Grandfather paradox)
Another thing that was mentioned was "dimension travel". Now... the use of the word "dimension" here is a bit misleading. In math, the word dimension has a very specific meaning. So I encourage people to not use the word "dimension" in the sense "traveling to another dimension". Instead, we can refer to them simply as alternate, parallel, or other universes (or sometimes "worlds").
Traveling to an alternate universe is not actually time travel, though it may seem like it. Imagine this: you build a house with two bedrooms next to each other and make them exactly identical. Every couch, chair, computer, etc... is exactly the same. You then proceed to live in one of the bedrooms for a month without cleaning much.
After the month, if you were to walk into the unused room, it would feel like time travel! Everything in the room would be like your room from a month ago! But obviously this is not time travel at all. The computer is not your computer, the bed is not your bed. It just so happens to look like the objects you're familiar to.
However, imagine if you did this experiment to someone who didn't know what was going on. What if you created a copy of their room, blindfolded them, and put them in the new room and told them you traveled back in time. They may just believe you. At least... the room would behave as if you had traveled back in time.
So travel into other universes isn't ACTUALLY time travel, but may wind up being the thing which is most plausible in terms of being even remotely possible.