I am a personal trainer, and I can tell you that there is a way to get stronger without getting much bigger. If you follow my advice, you will likely get a little bigger, but there is a definite difference between gaining muscle mass and gaining strength.
What a lot of people don't know is that there a lot of factors other than muscle size that affect a person's strength. For instance, the length of the muscle, the proportions of your limbs, where the tendons attach, and even the efficiency with which you are able to use your muscles.
To get strong, you must first ensure that you have proper form. I obviously can't demonstrate that here. If you are sure that you have proper form with most exercises, follow this advice: lift heavy weights. Don't do a ton of sets and a ton of reps. Pick weights that you can only do about 3-4 reps with. Sometimes you can do even fewer reps. Focus on compound movements like the bench press, squat, and deadlift which involve a number of muscles working in cooperation. Make sure that you do a few warm-up sets to get the blood flowing, but don't work yourself to exhaustion. It is more important, at least as far as strength is concerned, to do frequent workouts than it is to do extremely intense workouts that cripple you for days. The way most people work out will do very little for strength. Try doing a workout that involved a bench press, a squat, and a deadlift 4 times a week, with some other exercises like bicep curls, lat pulldowns, overhead presses, and calf raises thrown in. Do not work hard enough to make yourself sore with the main lifts. Just 2 or 3 sets with very heavy weights.
This may sound like a strange idea, but this method is very popular in eastern europe. A lot of the strongest powerlifters and olympic lifters have come out of eastern europe and have used these techniques. For much more comprehensive information, try to find the book Beyond Bodybuilding, by Pavel Tsatsouline. He knows his ****.