When exercising your legs, you can benefit from including both bilateral (two-leg) and unilateral (one-leg) exercises.
A bilateral movement has more potential for strength gains, as you are more balanced and can bring both legs to bear on moving the weight.
A unilateral movement can help even out strength differences between the legs. You can only bring one leg to bear, while the other helps provide stability or balance.
One excellent single-leg exercise is the single-leg split squat, better known as the Bulgarian Split Squat. It's not really clear if this originated in Bulgaria, or is merely credited to Bulgarian trainers and athletes.
Regardless of it's origins, it's a terrific exercise.
How do you do them?
Here are two ways. One is loaded with a barbell:
Barbell Bulgarian Split-Squat
Next is more common, the dumbbell version.
Dumbbell Bulgarian Split-Squat
Why do I recommend this exercise?
There are two things I like about this exercise.
It's Effective. This exercise is a personal favorite of mine and of Jungledoc, who writes The Road to Fitness. It's also a favorite of Joe DeFranco, who runs the gym I now train at.
"I'm a huge fan of single-leg movements like Bulgarian split squats and barbell reverse lunges. If you walk into my gym at any hour I guarantee you're going to see someone with their back leg on a bench, holding dumbbells to their sides or a barbell on their back. It sucks and it's hard as hell, but the weight just pours on." - Joe DeFranco, in "The Cure for Skinny" on T-Nation (contains NW/FS images).
It's Always Hard. I won't lie. Getting your first few reps of the Bulgarian-Split Squat isn't easy. You have to learn to position yourself far enough from the bench that you can touch your knee to the floor (or just graze it), but close enough that you don't lose your balance reaching back with the stabilizing foot. Once you do that, getting a lot of reps isn't easy. Weighting them with a pair of dumbbells makes them tricky, a barbell even more so. You can easily make them harder just by going heavier or adding more reps. Don't want to up the weight? Okay, try them:
...with the front foot on a step to extend the range-of-motion
...or try them with bands.
...or with chains.
...or with kettlebells hanging from bands.
Lots of exercises are amenable to variations. But this one is pretty hard right from the start. The variations are useful, but the basic version is sufficient to get a lot of useful one-legged strength quickly. You'll improve your balance after the first time you try them. From then on, it's just pure hard work.
Tip of the Day - If you do these on a non-padded or hard floor, put down a thin piece of matting or a rolled towel to cushion the knee. I've done these on a concrete floor using knee pads, but a rolled towel works just as well.