Loose Steering Components
If you've ever driven a new car and an old car back to back, you might notice that the steering in the new car is much firmer and more responsive than the old car. In other words, the newer car will respond more quickly to the way you turn the steering wheel, and the amount the car turns in relation to how much you turned the wheel should feel more accurate. (One caveat here: Different types of vehicles are deliberately engineered with different types of steering responsiveness, so this little exercise loses its significance if you're comparing, say, a sports car and a limousine, regardless of their ages.)
The point of this example is to explain that steering components, like many other parts on your car, can wear out, and since it happens so gradually you probably won't even notice. There are a lot of little moving parts that physically connect your steering wheel and the four wheels on the ground, and once those parts start to wear out, your wheels won't do exactly what you tell them to do. Your car will still steer (provided the parts aren't totally shot), but the excess play in that complicated network can cause vibrations.
These components are best left to the professionals, so keep this possibility in the back of your mind if you have an older car and the other potential solutions in this article prove fruitless.