There have been a couple good answers to this question already and many bad ones. I thought I'd throw my answer in there though.
Time exists. Even many of the people who have said time does not exist indirectly assume it exists. Yes, Einstein's theory of spacetime says that time and space, together, travel in two directions, like all dimensions, however, TIME still exists, it's just not (hypothetically) only going in one direction. Yes time may be relative, as proven with particles decaying longer in a particle accelerator than at rest (i.e. time slowing down), but then again, many other forms of measurement can be considered relative, such as distance when traveling close to the speed of light (your speed changes the distance to wherever you're traveling).
Another answer I didn't like is that time somehow exists through clocks. That's not really true, and it's misleading, because clocks are obviously man-made. TIME is not what's on your clock, however. Clocks merely perform an action at a regular interval (such as clicking the second hand at a regular interval, which clicks the minute hand 1/60th of the way towards the next notch of a minute and which clicks the hour hand 1/3600th towards the next hour number).
These terms, the terms that apply to clocks, "seconds" and "hours" and "minutes," "days" "years" "centuries" "millenniums" are all man-made terms and measurements of artificial durations of TIME. But they are measurements of something that exists.
Everything takes time. Every single event in this universe has a start and an end, and some duration of TIME in between. Nothing happens instantaneously. Time may be relative, but like I've said, so are other things that humans measure, such as distances, lengths, widths, etc.
Time does not go anywhere after it passes. It's a difficult concept to visualize, the fourth dimension, also known as spacetime. We humans can only experience life in the third dimension, and it is difficult for our brain to comprehend a concept it, well, cannot comprehend. In essence, if someone could see in the fourth dimension, they would see everything like it is in the present, like we see now, except with every single object, person, perhaps even down to every molecule and atom, the person would see everywhere it has been in the past and everywhere it will be in the future. Tough to think about, obviously, but once you can, you will realize that time does not "go anywhere," that spacetime that existed in the past still exists in the fourth dimension, and spacetime that is in the future does not get "created" but simply moves closer to the present, since it has always existed.
I hope that helps you understand your question a bit better. It's a very interesting question and one that promotes contemplation of some very difficult ideas, but it's a fun one to explore.