Which oil range you should use depends on where you live. The first figure is the low temperature viscosity of the oil (how thick it is at 0F/-18C) and the second is the high temperature viscosity (how thick it is at 210F/100C).
Essentially you want the first figure to be low and the second to be high, but if you live in a warm climate you can have a higher low number since your cold starts may not be THAT cold, but if you live in a cold climate you want to have the low number as low as possible. This can be important since if the oil gets too thick the oil pump can't handle it and you may either blow the pump or not get the oil into the bearings.
If you live in a very warm climate you may want to go for the upper range, like 20W50, in medium climate a 5W40 or 10W40 will do fine, and in cold areas a 0W30 or 0W40 oil is the best option.
It is important to recognize that even if the viscosity number is lower for low temperature the oil is still thinner at the high temperature.
Just watch out for single grade oils, you may find a bottle saying SAE50, and that's fine if you just do racing in warm weather (say NASCAR) since you don't do much cold starts, but it will be as thick as tar when you have it below freezing. Same goes for single-grade on the low-end, like if you find a SAE10 oil - DON'T PUT IT IN YOUR CAR - ESPECIALLY IF YOU HAVE A TURBO! You may use it in cases of extreme emergencies just to get home, but otherwise - use it in your lawnmower or lubricate the bicycle with it.
And when it comes to really cold weather I have even heard of people that drains the oil for the night and takes it inside to be able to start the engine in the morning. But this is for extremists.
And anyway - there is a graph in the owner's manual describing the recommended oil for each condition.
One other factor is also if you do a lot of cold starts and short driving. Then you may want the lower temperature figure to be a bit lower than normally specified since this will keep down the fuel consumption.
And when buying oil - there are also a set of other parameters to consider, viscosity is only how thick the oil is, but there are a set of API classifications too that are important. If you use an oil that doesn't fulfill the classification your engine needs you may ruin any chance of warranty.
Problems caused by "wrong" oil is usually caused by using an oil that fails the correct classification, and not really related to brand.
And finally - don't forget that some oils have additives that dissolves some rubber seals etc. This may be a problem with power steering and gearboxes and usually not engine oil, so watch out here! Wrong oil causing leaks can lead to costly repairs - even if the repair is to only replace a single seal.
Please notice that engine oils contains additives that makes it suitable for lubricating camshaft contact areas etc. This makes it unsuitable in gearboxes since it can mess up the synchronization in manual gearboxes or cause slippage in automatic gearboxes and require a gearbox rebuild (at least for auto).
You may also get some more information at these links: http://www.autoeducation.com/autoshop101/oil-change.htm