I love small engines, and fix them for the fun of it.  I've found that most of the problems with engines are pretty simple once you've seen them so I'm planning to document what I can to make "Fixin' Mowers" simple.  Here's what I've documented so far:

Doesn't Start:
If the mower was running just fine last Fall, and will not start in the Spring, the first thing to suspect is that the gas has gone bad.
If it just keeps running worse and worse until it stops (or doesn't start), it could easily be the air filter.
This is what a "Fouled Plug" looks like.
Tecumseh carbs are famous for gumming up and the mower may start but it will not keep running.  Here's a "quick" cleanup.
Here's "starts, but doesn't run"  problem from a very new Tecumseh with the mostly plastic carb.  The seal between the main jet and the carb bowl swells up and cuts off the gas.  I replaced it with a small O-ring.
You hit something and the motor will not restart.  Maybe it's a broken flywheel key.
B&S carbs with the carb on the gastank are VERY reliable, usually all they need is a new gasket between the carb and the tank.  While it's apart check the screens on the pickup - I've seen more than a few that have been plugged up.
No spark on an OLD (many before 1982) engine?  It's probably the points, but it may be easiest to replace the coil with a newer pointless model
Bring home a previously working riding mower and it doesn't even try to start?
WATER in the gas will stop a mower dead.  It goes to the bottom of the tank, and the bottom of the carb - where the engine pickes up gas.  Drain the carb and dry the tank if you find water in the carb.  This is water and gas in a jug, but if you see drops in the bottom of the carb bowl or tank, or see it bead up on top of the driveway or a rag, you have water.  Don't park the mower outside!
Flipped my mower over upside down, and now it doesn't start!

Doesn't Move:
Here's the normal problem on a shaft drive Lawn-Boy.
And here's an example of what happens when the belt drive clogs up with grass.  It happens with the Craftsman type horizontal belts, too.

Other Stuff:
Blowing white smoke out the exhaust may not be terminal.
Tecumseh carb floods or leaks gas.  It's only a matter of time until they do!.
If you let water sit in the carb, this happens.  You can clean it up, but it's a lot of hard, messy work!
It stopped for NO REASON, can be that grass has managed to gum up the drive.
Snapper Blade Clutch - Comet Blade Clutch
How does my Techumseh (most Craftsman) throttle linkage hook up?
How does my Briggs & Stratton throttle linkage hook up?  (the older mowers only have the larger spring)

Big Problems:
You can split a late model OMC Lawn-Boy case with the engine on the deck.
You hit something and the motor will not restart.  Maybe it's a broken flywheel key.
If it's not the flywheel key, it could be one of several PLASTIC gears  inside the motor.  This example is the govenor gear after a quick stop.
Did your 5-6hp Techumseh BLOW UP for no good reason?  It wasn't YOUR fault, they ALL do that!

Time, time with him was time in school
Sitting in the backyard
Teaching from his box of rusty tools
And he could talk such magic at his clear-eyed 85
And just when you thought you got the picture
he´d show you the joke inside.
"Yeah", he said, "The joke´s on you!"
"You don´t even know where you´re going to!"
                                 John Sebastian - "The Four of Us"