Type 1 - Only the Audience is Transported
In this kind of time jump, you are essentially just hitting the fast forward button on the world. Life goes on 'naturally' for everyone inside the world but a large amount of time is skipped. Whether it's a few weeks, a few months, or a few years the effect can be the same. The audience stops watching at a point, and then is reintroduced to the world after time has passed. Generally, they are not told what they missed right away, but instead are reintroduced to the main characters, to the new situation in the world. Often this is used to gloss over times where things are less interesting, quickly getting to the next bit of interesting story.
Another use is progressing through a character's life, letting you go from when they were young and inexperienced to them when they're in their prime and have developed much further into their potential. This lets you go from the character being bumbling to significantly more powerful. With this growth in ability can also come their maturity, leaving up to mystery (for the time being anyhow) what happened to them to make them develop in that way, assuming of course that they are not just like they always were.
Yet another way, is how DC used this fairly recently. You move everything forward, and have momentous occasions happen in that time period. You then slowly reveal what actually happened over the course of the story, while the audience, and characters, are attempting to deal with the tumultuous changes that skipped time brought. As said, DC did this with their recent "1 year later" leading into the hit series 52.
Type 2 - 'Only' the Character(s) is/are Transported
Only is in ' ' for this because with the character being transported, the audience is as well. However, the point is that the character is moved far into the future or past, sometimes even to another world. With this they get to see how things, based on a certain course of action, will play out to their extreme end. Alternatively, they may see how the story begins. Either way, they can get a new perspective on things, the origin of an antagonist, or just what they are fight for/against.
Another use of this method is simply to change the setting of the story by vastly changing the time frame. This is usually followed with a number of awkward moments as the person has to learn to cope with the differences in the time period. For people moving forward in time, it's getting used to the new technology. For backwards in time, it can be the backwardness of civilization, the violence, or other things like the lack of hygiene. Not that these things aren't without their merit for humor, but do try to approach things thoughtfully. That being said, the drastic changes in the world can be shocking in and of itself as well, seeing the world the hero loved gone to ruin because no one was there to stand against the villain can get the point across. For those of you who play the Zelda games, do you remember the first time you pulled the Master Sword in Ocarina of Time? Your first view after leaving the temple? It can be a powerful experience, especially for a video game. Use it wisely in your story if you do put it in.
Using Type 1
One of the better ways to use Type 1, be it for a book, a comic, or a game, is to let the players/audience 'grow' with the character. How you do this is simple enough, you start very near the beginning for the character and you show how they come into the life of adventure/intrigue that they are in. Go through what training they get (if any), their early adventures, and them being the small fry with potential in the big scary world. Then skip some time, a few months, a few years, it doesn't matter. For a game you're going to want to give the players some extra XP to let them grow the character and talk with them about what they are doing over that time. When you pick the story back up, the character is older and they aren't the small fry (or at least, they are somewhat bigger), they've grown and developed as a character and are stronger. The adventure they go on will be a bit grander in scope, a bit more mature. This, as the second act, is a good time to show that the world is not all fun and games, and that people can and do get hurt in the type of life they are leading. Afterwards, you do another jump (hand out more XP if this is a game) come back and show a bit more.
What does this all accomplish? For one thing, you cover a significant portion of your characters life in a significantly faster time than normal. This allows you to tell stories that have a meta-plot of years, as well as to show your hero(es) growing up from young 'children' to full adults. The impact can be amazing, as you focus on the truly interesting and character defining moments, and then let the slower times pass quickly. The audience is privy to the characters life, they get to see their hopes, dreams, and aspirations change as they grow older. Goals set, and then achieved, failed, or passed on as time moves on. If the story kills the character, it can have more impact because this is someone that the audience has known since they were 'very young'.
Finally, the time jumps let you set plot items into motion quicker. You can just say things happen in the time you are going over, explain it yes, but it doesn't need to be detailed because that the plot was set in motion is less important than the fact that it was set in motion. It lets you bring up new events quickly, without having to 'waste' time between acts showing the set up for things, especially as those set up acts can be incredibly boring to see happen in greater detail.
Using Type 2
Type 2, as I said above, is best used as a setting change tool, or for giving the character a chance to see the beginning or the consequences of some action. For a table top game I don't recommend showing them the beginning, unless the plan is to change the past, because players will always find a way to mess things up real good if you let them loose in the past.
Moving things to the far future however can be good. It can give the character a chance to see just what they're fighting for. That without them, the world actually is doomed. There is no one else to stand up to this evil, because if the person who was supposed to but gave up half way did that, why should they try that hard? Or even better, their friends did try without them, and died or worse for their efforts.
Alternatively, the time jump can be the entire point of the story. You introduce your characters, your world, and the going ons. You let the players get to know them, and then you drastically change the setting via a time jump or other manner. We then get to see these characters reacting to a completely different set of rules, we get to see what they're made of as they deal with this situation. Perhaps someone from their time doesn't have a control chip in them, and so can fight back against the technological overlords. Or they're just more stubborn. Perhaps some 'near future' ingenuity and knowledge is all that is needed to avert disaster. The reason they're special can be anything, or nothing, the fact is that to find out what a character or person is made of you need to move them out of their comfort zone, and a drastic change in setting does that quite nicely. You are almost literally standing the world on its head on them.
No matter what kind of story you are telling, or the medium you are using to tell it, Time Jumps are a powerful tool that you should consider using. At the very least you want to understand what they can do, so that your not using them is a conscious choice, and not from a lack of knowledge of the tool available to you.
For a role playing game, this tool is just as strong if not stronger. It lets you do a lot of things very quickly, such as summing up events between what you want to actually run, and moving the characters up in power level with a natural reason present for it. The ability to also let your players have their character through the majority of their character's life is also something you can't match in many other ways.
If you haven't tried using one before, you should give it some serious consideration to test it on for size.