The two common problems when troubleshooting a Briggs & Stratton lawnmower are that the engine doesn't run, or the mower doesn't cut.
If the engine doesn't run, there are either one of two things wrong, no fuel is getting into the cylinder to be ignited, or there is no spark to ignite the fuel once it is in the cylinder.
If it doesn't cut, there is something amiss with either the blade or the driveshaft.
You'll need fresh gas at all times. If your gas is a season old, it needs to be drained and replaced with fresh fuel. Try starting it with fresh gas after the carburettor has been primed. That may fix the problem.
If you have fresh gas, prime the carburettor several times and pull the cord a few times. The mower should at least fire or 'pop' as they say, and if it doesn't, that generally means that the spark plug is bad. It will need to be replaced, and should ideally be replaced once per year.
Other things to check are fuel shut off valves, which if turned off won't allow any fuel to get to the carburettor; the throttle, which needs to move into the choke position initially or your mower won't start; and the emergency shut off handle which needs to be pulled and held on the main handle, or the mower will not start. Lastly, check the wire on the spark plug to make sure it is secure and tight.
Blade and driveshaft
If the mower is cutting erratically or if there's a terrible vibration when cutting, it usually means that either the blade has been damaged or the driveshaft may be compromised.
In many cases rocks or other objects can damage or bend the blade, causing it to go out of balance. You can attempt to file or grind down the blade yourself and attempt to get it back into balance, but perhaps the easiest fix is to replace the blade with a new one.
Make sure the centre nut holding the blade in place is tight, as well as the bolts holding the engine to the deck.
If vibration persists after a new blade has been correctly installed, chances are the blade driveshaft has been bent, which requires professional repair.