For desktops, it's debatable which one is better. See below. For laptops, shutting down is most likely the best option.
The argument for shutting down is that computer parts have a limited lifespan, say 30,000 hours (just an example, I have no clue what it actually is), and that leaving the computer on uses up the lifespan of its components. By shutting down, you'd be saving the parts from unnecessary wear and tear, and you should be able to keep your computer for longer. Also, shutting down saves energy, for those of you paying the power bills. IIRC, 1 computer with 17" CRT monitor uses about $150 of electricity a year, and shutting down your computer obviously saves some or most of that money.
The argument for keeping it on is that more wear and tear occurs when starting up than during idle periods. For example, on startup, the hard drive has to start spinning, putting strain on the motor, and decreasing the lifespan of the components.
Both of these arguments don't really apply to desktops anymore, since most parts have long lifespans that will likely outlast the usefulness of the computer (for most people, at least). So for desktops, I would recommend shutting down, because it doesn't affect lifespan enough to matter for most users, but shutting down saves energy (and who doesn't want to save energy
For laptops, it's a bit of a different story. Laptops are essentially desktops packed into a little case, with more restricted airflow and more heat. Leaving my computer on overnight makes it very warm (I have no exact temps, but certainly hotter than turning it off). Heat is something that will hurt the lifespan of computer components. Therefore, my position is to also shut down laptops at night. This gives them a chance to cool down before being used again. Wear and tear from starting up every day is minimal compared to the wear and tear from excessive heat. Even a laptop with big fans is going to get hot, it's just what happens.
Final verdict: shut down. For desktops and laptops. Save energy and save your computer from heat.
Hope that helps