This can be caused by several things. First thing is to check your tyres and wheesl are balanced as a severely out of balance wheels will show up as vibration under braking.
If the steering wheel vibration or steering wheel shake as it is sometimes called happens when driving the car without braking it is without doubt a balance issue. The vibration will often occur at a certain speed and driving at speeds above or below that the vibration may disappear.
It can be as simple as the balance weights becoming unattached and may cost a few dollars only to get the wheels balanced. Vibration can also occur as a tyre wears unevenly.
Steering wheel vibration under braking is a little more expensive to fix. It is usually associated with the brakes on your vehicle or steering geometry, both of which are linked.
If you have recently had new brakes fitted a vibration may be noticed even if the car had no vibration with the older brakes. Drivers often chastise the fitter or the brake rotor supplier but it is rarely them that are at fault.
It is without doubt that brake pads and brake rotors have to be fitted properly, especially brake rotors and that all the standard precautions such as clean and preparing the wheel hubs done,checking for hub run out and tightening the wheel nuts or lug nuts sequentially and finishing them with a torque gun but in some cases when all this is done perfectly vibration appears after 2000-4000 miles. This is a clear indicator that integral run out of the vehicle is the culprit.
One in seven cars suffer from geometry issues. These can have been in the car from new and it is interesting to note that many car manufacturers actually machine the rotors on the car to align them to the car. Subsequent replacement with a new set of brake discs or brake rotors will result in steering wheel vibration unless the rotors are re-aligned on what is called an "on car brake lathe". These re align your rotors and the vibration will be gone.
Some drivers will of course resist this cost but it is actually the only way the issue will be solved. It is interesting to note that all high end car dealers such as Porsche,Jaguar and Honda plus many more all possess such on car brake lathes for this very reason.
Geometry can also be upset by road pothole and one interesting article here is from a car dealer who offers to even check customers cars after the winter pothole damage in roads. See article here on pothole.
Steering wheel vibration is not to be ignored, it can upset braking and upset your willingness to actually brake your vehicle as much as being annoying.