Riding Mower Help Needed

I have a Murray riding mower that normally works fine. I even had it over-hauled this spring. The other day I went to start it and it started to turn then nothing. I thought I'd let the battery go dead, so I put the battery on the charger. Charger showed it as full not long after putting it on. I put it back in the mower - nothing. My neighbor and his friend came over to see if he could help. He tried to start it using a portable battery jumper. Would not jump from the battery. They jumped it directly from the engine (which I thought was very scary, and is probably something dangerous.....) by attaching one cable to a rod looking protrusion coming from the engine and then tapping something else (it all happened so fast). It started and ran, and I mowed fine. Of course once I stopped it, it still wouldn't start. I can't figure out what's wrong, and I really hate to call out for service and pay the $$$ if it's something simple.

I have very little mechanical skills (though I'm trying to learn), but I like to tinker if I have something to go on. Anyone have any ideas?