Cars are complex mechanical devices with hundreds or thousands of parts that can go wrong. But just being out on the road, being aware of your car can give you hints as to what may be wrong. Before checking any of these things, always stop the car somewhere safe and turn the engine off. All of my suggestions are from my personal experience and/or knowledge. If you damage your car or yourself as a result of trying anything, I'm not being held responsible.
Minor wobbling of the steering wheel
All car tyres are balanced. This is because the rubber in the tyre isn't 100% uniform, and there's also the weight of the valve to take into account. When a new tyre is fitted to a wheel rim, the mechanic should balance it. This is done on a special machine that spins the tyre and measures its wobble, then tells the mechanic to clip one or more little weights, typically between 5g and 100g, either to the rim of the wheel or (more commonly on alloy wheels) inside the rim. These can get knocked off or fall off. It's always worth checking. Driving a short distance with a badly balanced tyre won't do any harm, but get to a garage. I suspect most tyre repair places would balance a tyre for (almost) nothing.
More serious wobbling of the steering wheel
The tracking may be off. This is the alignment of the wheels with respect to each other. This needs to be sorted by a garage. Again, driving a short distance won't do any harm, but in time, you can get uneven tyre wear with badly set tracking. Hitting a curb or of the like can knock off the tracking.
A wheel may also be loose. Check all the nuts are present on all the wheels. If you have recently changed a tyre, check that one particularly. If put on correctly, the nuts shouldn't loosen themselves!
Very serious wobbling, together with noise
This often indicates a flat tyre. If you have a front wheel drive car, and the flat is at the back, it can be hard to tell without listening, as the car will appear to drive almost normally. Stop and check IMMEDIATELY - driving on a flat tyre, apart from being dangerous, can often damage the tyre beyond repair as parts of the rubber that aren't supposed to be in contact with the road and wheel, are. Change the tyre and get the spare fixed.
Squeaking from the engine
This often indicates one of the belts in the engine is slipping. The "Fan belt" is the most well known, although in most modern cars with electronic thermostat controlled engine fans, it doesn't actually drive the fan. Never-the-less, there's always a belt from the engine to the alternator, and for cars fitted with them, belts to drive the power steering and air conditioning compressor. The belts should give slightly when prodded (ALWAYS STOP THE ENGINE BEFORE DOING THIS!!!!) and should have clearly defined "teeth" on the inside.
There is also a timing belt in car engines. If this goes, you have a major (expensive) problem. But this should be replaced as part of the regular service cycle.
Engine sounds stuttery
This is most likely due to a cylinder not firing. Often this is caused by one of the spark leads not being pushed in properly. It could also be a dodgy spark plug but this is something you can't easily check by the side of the road. On older cars with a manual distributor, the distributor cap may be incorrectly adjusted so the rotor isn't touching the points correctly.
Loud noise from engine
People don't realise just how noisy an engine is, and what a good job the exhaust system does of silencing it. If you suddenly get a loud rumble (often accompanied by vibration), there may be an exhaust problem. The system consists of a number of sections connected together, and sometimes one of the joins fails. You probably need to take the car to a garage to get it seen to.
Sudden engine overheating accompanied by a plume of steam (courtesy of pjd)
Often this is the Head gasket going. Essentially the gasket is a thin sheet of cork, rubber or metal that goes in between the 'head' of the engine and the block. If you run the engine too hard (over revving) the head can become too hot and warp. This means that gasses from the piston can escape, usually into the coolant system (which causes the engine to overheat). The head gasket can also persish over time and cracks form, allowing some leaking. Other symptoms include rough engine running/mis-firing as gas escaping (and coolant entering) the piston interferes with the firing. Bubbling in the radiator tank is a sign too. There can also be some problems with the fuel mix being wrong which contributes to general engine mis-behaviour. Be prepared for a large repair bill!
If anybody thinks of anything else, please let me know.