An oily spark plug on a lawn mower can be caused by a number of issues, ranging from minor mechanical problems to serious engine damage. In either case, the spark plug will need to be cleaned or replaced and the spark plug port will need to be cleaned of oil before the lawn mower engine will start.
Over Filled Oil
If a lawn mower engine is filled with more oil than the manual indicates is necessary, the crank case can overflow and cause oil to leak onto the spark plug. The solution is somewhat self explanatory. Remove the oil cap and turn the lawn mower over to pour out some of the oil. Recheck the level to make sure you have not removed too much, and add oil back as necessary.
One potential source of oil on your spark plug is a blown piston seal. The seals keep oil from leaking into the combustion chamber and, if blown, can manifest themselves by spraying oil on the spark plug. This is not a user-serviceable part, as it requires an engine rebuild. Contact your local authorized dealership for your particular make and model lawn mower.
Cleaning the Plug
If your spark plug is otherwise new, you may want to attempt to salvage your old plug rather than replacing it. You can clean your spark plug using carburetor cleaner or brake cleaner, available from any automotive store. Spray the plug down with the cleaner and wipe it off with a shop rag.
Cleaning the Spark Plug Port
In order to get your mower's engine to turn over, you are going to need to clean out the spark plug port as well. Any residual oil may prevent the spark from igniting fuel and keep the engine from starting. Push a rag in to the spark plug port and attempt to wipe out as much oil as you can. Spray the port with starter fluid, and wipe it once more thoroughly. Re-install the spark plug and start the engine.