This is one of those questions where there is no single right answer. In other words, it depends on how you use your computer.
There are at least three situations that force you to leave your computer on 24 hours a day:

• You are on a network, and the network administrators back up files and/or upgrade software over the network at night. If that is the case, and you want your machine backed up or upgraded, then you need to leave it on all the time.
• You are using your machine as some sort of server. For example, HowStuffWorks has a machine that creates the images for the How Webcams Work article. It needs to be on 24 hours a day. If your machine acts as a file server, print server, Web server, etc., on a LAN (local area network) or the Internet, then you need to leave it on all the time.
• If you are running something like SETI@home and you want to produce as many result sets as possible, you need to leave your machine on all the time.

If you do not fall into any of these categories, then you have a choice about whether or not to leave your machine on. One reason why you might want to turn it off is economic. A typical PC consumes something like 300 watts. Let's assume that you use your PC for four hours every day, so the other 20 hours it is on would be wasted energy. If electricity costs 10 cents per kilowatt-hour in your area, then that 20 hours represents 60 cents a day. Sixty cents a day adds up to $219 per year.
It's possible to use the energy-saving features build into modern machines and cut that figure in half. For example, you can have the monitor and hard disk power down automatically when not in use. You'll still be wasting $100 per year.
The argument for leaving your computer on all the time is that turning it on and off somehow stresses the computer's components. For example, when the CPU chip is running, it can get quite hot, and when you turn the machine off it cools back down. The expansion and contraction from the heat probably has some effect on the solder joints holding the chip in place, and on the micro-fine details on the chip itself. But here are three ways to look at that:

• If it were a significant problem, then machines would be failing all the time. In fact, hardware is very reliable (software is a whole different story, and there is a lot to be said for rebooting every day).
• I don't know a single person who leaves the TV on 24 hours a day. TVs contain many of the same components that computers do. TVs certainly have no problems being cycled on and off.
• Most vendors will sell you a three-year full-replacement warrantee for about $150. If you are worried about it, spend some of the money you are saving by turning your machine off and buy a service contract. Over three years, you come out way ahead!