First off here's a closed guard:

What makes it 'closed' is how the guy on the bottom has his legs closed around his opponent. That makes it harder to pass into a better position.
If the legs aren't closed in this position, it's an open guard:

The guy on top here has also postured up, allowing him to pass easier and deliver more forceful strikes.
If you're on your back and don't know what to do, getting into closed guard and holding your opponent's head down to keep him from posturing up is your best bet. After a while, the ref should stand you up.
Guys more trained in jiu-jitsu are able to do a rubber guard:

Similar to an open guard, but grabbing the ankle with the hand and bringing the leg over the person's back prevents them from posturing up, and it also opens up a few transitions and submissions.
If you're in someone's guard, the most basic pass is into half-guard:

Instead of having both legs inside an opponent's guard, now there's only one leg. Getting that leg out makes it easier to strike, and provides better passing opportunities. Sometimes you can pass right to mount, but more often than not you'll pass to side control:

In side control, it's easy to control your opponent while still doing good damage with elbows and knees.
If you want, from here you can move to north/south:

Not very popular since it's tougher to strike in this position than in side control, and the only thing it opens up is the rather difficult north/south choke that only the best jiu-jitsu guys can usually pull off.
If you're looking to improve off of side control, your best bet is to try and get into a full mount:

If you can posture up in full mount, you can end a fight quickly.
If you've been mounted, you can either try and pull the head down and stall, hoping for a stand-up, or you can roll and give up back control:

Notice how the guy with back control has both feet inside his opponent's thighs. That's called having two hooks in. The hooks make it easier to control an opponent.
Alternatively, you can get back control and have a body triangle:

Getting that foot under the knee and squeezing makes it very tough to get out of.
If you give up back control, the best bet is usually to try and roll over and end up in the other guy's closed guard. And the ground game begins anew.
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