When was the last time I changed the oil in my mower?

Orientation isn't terribly relevant, although it is easier to design a two stroke engine in an "odd" orientation, because it does not have to be concerned about engine oil draining back into a sump.

In the simplest kind of two stroke, the compression and intake happen at the same time. Piston goes up (toward the cylinder head), compressing the fuel in the combustion chamber, while at the same time, fuel is drawn in beneath the piston, around the crankshaft. At top dead center (ish), spark occurs. Combustion powers the piston down (power), while at the same time forcing the fuel charge from beneath the piston into the combustion chamber via a transfer port, which also directs the exhaust to exit the exhaust port.

These two strokes - one up, one down - are one complete "cycle" of the engine's operation.

In a four stroke, starting with the intake stroke, the piston moves down and the intake valve opens via a camshaft. Intake fuel charge enters the cylinder between the piston and cylinder head. The next stroke is compression. Both intake and exhaust valves are closed as the piston goes up. When the piston is all the way up (ish) to the cylinder head, spark occurs, causing combustion. The intake and exhaust valves remain closed, and the piston moves down. That was the power stroke. The next time the piston moves up, the exhaust valve is open, and exhaust is ejected through it. Intake, compression, power, exhaust; or colloquially "suck, squeeze, bang, blow." Those four strokes complete one "cycle" of the engine's operation.

Each engine has a crankshaft. This is the rotating shaft on which the piston is attached with a connecting rod. On the crank end of that rod there are bearings. (Two strokes will use roller bearings while four strokes will (usually) use flat bearings.) The connecting rod attaches to the piston with a wrist pin bearing. These bearings require lubrication. Without a thin layer of oil between the metal surfaces moving against one another, friction will quickly create heat and catastrophic bearing failure. (A four stroke also needs lubrication of the valve train, contained in the cylinder head; two strokes have no valves, and so no top end lubrication is required.)

A four stroke engine accomplishes this lubrication with thick engine oil. Most commonly, this oil collects in a "wet sump" oil pan at the bottom of the engine, is picked up by an oil pump, and circulated through the engine, being directed at the parts which require it, finally draining back into the sump. (There are also "dry sump" systems, where the oil resides in an elevated oil tank.)

A two stroke engine - since the crankshaft portion is also the engine's fuel intake - does not have an independent oiling system. Such a system would interfere with the fuel intake, so the lubrication for the crankshaft bearings has to be included in the fuel. As described previously, this can be either with an oil injection system, where oil in a separate reservoir is delivered directly into the crankcase, or by premixing thin two stroke oil into the gasoline. In either case, the two stroke oil provides lubrication to the crankshaft, and then is burned in combustion. This is why two stroke engines have tha blue exhaust plume; that's the two stroke oil burning.

Two strokes also have incredibly good power to weight ratios, mainly because they have twice as many power strokes per engine rotation. They are also incredibly simple, what with not having camshaft and valve train. Why don't we use two strokes all the time then?

Well, mainly because they burn oil by design. But they are also noisy, and their design makes them severly RPM limited (they'll lose power as they're unable to intake enough air/fuel to keep running). Where two stroke engines shine is in very small applications (weed trimmers and model airplanes) or very large applications (industrial and maritime), though the latter makes use of fuel injection as opposed to carburetion, and does use a wet/dry sump engine lubrication system.