How many reps?
A "rep range" is the number of repetitions you do in a given exercise.
The most common rep range you'll find people doing is sets of 10 repetitions. I have no numbers on how common it is, but if you ask all the gym-goers you know what they do, don't be surprised if they say "sets of 10 reps" or "sets of 8-12 reps."
This is often regardless of their goals.
"I want to be stronger." - probably doing sets of 10.
"I want to lose weight." - probably doing sets of 10.
"I need more endurance." - maybe sets of 10?
"I want to get big." - Oh, perhaps sets of 10.
This is a very common recommendation. You'll find it on EXRX (quoted from the ACSM). You'll find it in most diet-centered workout books. You'll hear it on the web all sorts of places. The only real variations you'll come across in popular media is 12-15 reps, usually recommended for women. Women, its said, need the higher reps to "tone" - although "tone" is poorly defined at best.
Not that there is something wrong with 10 rep sets. The problem is that it's usually fundamentally divorced from the goals of the lifter. It's given as a blanket recommendation to all lifters. What rep range you do should depend on the exercise you're doing, and why you are doing it.
So let's look at various rep ranges.
Your body responds to weight training by a process of supercompensation. Basically, it gets better at doing the things you do. It's supercompensation and not just compensation because your body over-responds to the stimulus. It will prepare for the next incidence of that stimulus (a given weight, number of repetitions, etc.) by overcompensating slightly. So this time 10 reps of 135 pounds is hard, but next time, your body wants to be prepared to do a little more. How it supercompensates is based on the stimulus.
Low Reps (1-5 reps) : Generally, exercising for low reps emphasizes muscular strength and neurological improvements. That is, your body gets physically stronger, and learns to activate your muscle fibers more efficiently to let you lift more. Each repetition is so close to the body's maximal strength that it reacts by improving that maximal strength for next time.
Short Version: Your gains are more strength than size or endurance.
Medium Reps (6-12 reps) : When you lift for medium reps, you get some strength, but not as much. Your body responds by increasing muscular size, primarily. It doesn't get so much stress that it needs to re-pattern itself to fire more muscle fibers faster, like it does at low loads. But it does get stressed in a way that requires more muscular fiber size and more endurance, so it improves them. It doesn't need so much strength, so it doesn't improve that as much.
Short version: Your gains are more size and endurance than strength.
High Reps (13-20+ reps): When you lift for high reps, you get little strength, but more endurance. Each repetition doesn't require that much strength, so your body doesn't require better neurological efficiency or muscular size. It does require more endurance, because you are asking it to work over and over again.
Short version: Your gains are more endurance than size or strength.
Very High Reps: At this level of repetition, it's all endurance. You are lifting a weight or doing an exercise that doesn't need more than a fraction of your strength. You begin to limit yourself only by endurance - you run out of energy rather than run out of muscular strength needed to keep lifting. If you think about it, a marathon is nothing but 26.2 miles of jogging steps, each of which is a single rep. You are so far into the endurance continuum that your body may sacrifice size and strength, which is why marathon runners don't look like sprinters, although both run.
Short version: Your gains are all endurance.
All of these assume you are training with an appropriate weight. That is, you are doing 1-rep sets with something at or very close to your 1-rep maximum, or that you are doing 10 reps with your 10-rep maximum. The number of sets matters - if you do 1 set of 10, you can probably go a little heavier than 3 sets of 10, but in the end the difference isn't important to the discussion at hand.
All of this is on a continuum. A set of 6 repetitions has a bit more in common with 5 or 7 reps than it does with 1 rep or 11 reps.
Here are two excellent articles that discuss this in more depth.
The first is an excerpt from the first edition of Starting Strength. It's the best chart available on rep ranges. As you can see, the lower the rep range, the better for strength.
(Edit: This link is no longer valid, however, this chart covers the same ground, and it's by the same authors.)
The second is a web article by Tom Venuto. It also goes into depth on the subject, and I recommend reading it through.
What Happens within the muscles in response to different rep ranges?
So what range should I use?
That depends on your goals.
Strength athletes like powerlifters lift for low reps because they need maximal strength to succeed in their sport. They are not judged on endurance or size, but on maximal strength.
Bodybuilders generally train in the 6-12 rep range, often 10 reps, because it promotes hypertrophy (muscle size increase) the best. They succeed in their sport by demonstrating size and muscle attractiveness, and are not called on to demonstrate strength or endurance.
And so on.
This isn't cut and dried - some systems mix up rep ranges. For example, Conjugate Periodization uses several rep ranges together. You might do back squats for 5 sets of 3 reps, but then do glute-ham raises for 4 sets of 8 and then reverse hypers for 4 sets of 10. The Westside for Skinny Bastards routine mixes several rep ranges, aiming to develop maximal strength, size, and endurance all together.
What's common to all approaches, though, is that they only work if they fit your goals. If you need strength without size but train for size and endurance, you're not training to you goal. If you need endurance but increase only your maximal strength, you aren't training to your goal.
I'm just starting out, what should I do? You can try a variety of rep ranges. But if you are looking for size and strength together, your best bet is to start with sets of 5 reps. This is the basis for the Starting Strength routine, for Bill Starr's 5x5 routine, and for Stronglifts 5x5. The Power to the People routine is all sets of 5 reps. Not all beginning programs use 5 reps, but those that do have a strong reputation for success. They straddle a nice middle ground between strength gains and gains in muscular size in a way that complements each element.
So, if I can do 10 reps at 135 pounds and I do 1 rep instead, it's better for strength? No. The rep ranges in those two articles assume you're really only able to lift that weight correctly that many times. Lowering the reps won't make a difference unless you raise the weight to match your new rep range.
Are low reps more dangerous than high reps? No, they really aren't. You aren't any more at risk doing a 5-rep maximum weight for 5 reps than you are doing a 10-rep maximum weight for 10 reps. The weight you lift is lower, yes, but each of them represents the maximum number of times you can lift that weight. Beyond that (the 6th or 11th rep) is not possible, therefore the final rep of each is equally dangerous. Both represent a maximal effort.
To put it another way, if lower rep were less dangerous, you wouldn't ever have repetitive motion injuries - people don't get carpal tunnel from lifting heavy weights but from typing or moving their arm in the same pattern to often. Tennis elbow and golfer's elbow aren't from massive weight stress but from a low load done too often.
What's the best rep range? By now, you should know the answer is, "for what?" It depends on your goals. If you've been working in a single rep range for a long time, the answer usually is "Whatever range you aren't doing now." The suggestions can be a little extreme but effective. If you're doing 3 sets of 10 and you're stuck, try 10 sets of 3 (at the appropriate weight for triples). If you're doing 5 sets of 5 reps and stuck, try 10 sets of 1 rep or maybe 3 sets of 10-15 for a while, and see if that breaks your plateau. But in general, beginners should probably stick with a rep range like 5 reps and just get stronger...all beginners have the same problem, they need more strength!