Hi Jim
That little rubber button is the primer. It shoots in a few mls of petrol into the carburettor to start the engine. If it has lost its suction, the rubber is broken and needs to be replaced. can be got readilly at any lawnmower repairer. Easy to replace also. When you get a new one you will know better how to replace it.
If you need to keep it going till then you can always take off the air filter ( usually 1 screw through it) and put in some petrol straight into the carburettor. (Only about a teaspoon full). Then start engine and replace air filter. Hope this helps.
Diarmuid Horgan, June 2007