Most walk-behind and all riding lawn mowers have 4-cycle engines -- you add oil to these separately from the fuel. This oil is supposed to remain in the crankcase; if any gets into the combustion chamber, it burns with characteristically blue smoke. Smoke of this color doesn't necessarily suggest immediate problems with your lawn mower, and it may go away. If your mower is old, though, blue smoke could mean that internal seals are worn and need replacing.
Nothing to Worry About
You may be puzzled because your brand-new lawnmower is emitting blue smoke, but the reason is usually benign and easy to explain. If you use your lawnmower on a hill with a slope greater than 15 degrees, oil can pass into the combustion chamber through the air intake port, and the mower will exhaust puffs of blue smoke. They should go away when you return to level ground. The same thing can happen if you overfill the crankcase with oil or turn the mower on its side to service it. If you have to turn the mower on its side, do it in such a way that the spark plug is facing up to avoid this problem.
Leaking Breather Tube
Many lawn mowers have a breather tube that extends from the combustion chamber to the air intake port and muffler, and this tube has a extension that collects oil spray from the crankcase and directs it back into the crankcase. If this tube leaks air, oil can spray into the combustion chamber and burn. As a side effect, the breather tube directs oil to the air filter, which gets clogged and can't do its job, and then the engine may begin performing poorly. The problem is easy to diagnose by examining the air filter for oil fouling.
A Sign of Engine Wear
The purpose of oil is to lubricate the moving parts of the engine, including the pistons and valves, and these engine parts have seals to prevent oil from bypassing them and getting into the combustion chamber. In addition, a head gasket around the entire valve assembly separates it from the crankcase. It's normal for these seals and gaskets to wear over time; eventually small nicks develop, oil passes through and the engine begins to emit blue smoke. The amount of smoke that the engine exhausts indicates the amount of wear -- excessive smoking is a sign that you need to replace one or more of these seals.
What to Do About Blue Smoke
If you see puffs of blue smoke while you're operating your mower on a hill, simply walk or ride the mower back to level ground. If the puffs occur when you're mowing on level ground, check the crankcase oil level and drain any excess you find. The oil level shouldn't exceed the highest notch on the dipstick. Correcting blue smoke exhaust from an older walk-behind or riding mower may involve some engine disassembly so you can replace worn gaskets or seals. This is a job for the mechanically inclined -- if that doesn't describe you, take your older lawn mower in for servicing.