The FEI governs all international show jumping, and most countries try to keep their rules in line with the FEI standard, although some may have some variations at the lower levels.
The basics outlined above are correct - don't start until you hear the bell, if you have a stop and demolish the jump, it will be rebuilt and you will wait for the bell to signal you to start again - although the scoring outlined above is not in line with FEI rules. In the past a refusal was 3 faults, however now it is four, and a second refusal results in elimination. In some countries, I believe the lower levels may still have a three refusal elimination rule. In the US it is a two refusal elimination all the way down to the smallest jumper classes.
Show jumping is scored on the basis of faults you accrue while competing. Faults are earned over the course, by lowering the height of the jump by knocking a rail, or by stopping or running out of a jump. You will also be scored as having refused if you deviate from your path in a way that you "cross your track", such as making a circle. Faults can also be earned on the clock, by going over the time allowed.
Jumper classes can actually be held over a number of different formats, but it's most common to have a jump off, so I'll discuss that format.
Prior to entering the ring, you will learn and walk your course to make a plan for how you will ride it - regarding what turns you will make, how many strides between jumps, etc. At this time you will also see what the time allowed for the course is.
Once you enter the ring with your horse you will hear a bell sound - in FEI rules that means you have 45 seconds to get to the first jump. Don't start before you hear the bell!
If you jump clean over the first course - that is, no rails, no stops and within the time allowed - you will progress to the jump off. In the jump off you want to jump clean AND be the fastest. In the first round you just want to jump clean and within the time allowed - your time doesn't matter if you are clean, as you will jump off subsequently.
If your horse stops or runs out on course, but doesn't dismantle the jump, the time will keep running and you will be allowed to try that jump again. If your horse runs out in the second or third part of a combination, such as ducking out of the second jump in an in-and-out, you will have to jump the whole combination again. Combinations are considered one obstacle and you can tell this because the jumps will share a number, 5a, 5b and so one whereas as jumps not in a combination will have individual numbers. If your horse stops close enough to the jump that he knocks it down, the clock will stop and the jump will be rebuilt. You will then wait for the bell again and readdress that jump once you hear the bell.
I think that covers the most basic rules! I do not compete in eventing, so I'm not familiar enough with their rules to give a good outline.