I echo jzd's answer. The car's owner's manual should tell you what you need.
To answer some more of your question: numbers like "5W30" are viscosity ratings. They indicate how viscous ("thick") the oil is. Most oils are "multi-viscosity," quoting a range of weights (e.g., 5W30 instead of 30W) to indicate how they behave at different temperatures. The "W" stands for "Winter" and indicates how the oil performs at cold temperatures. For example, 10W30 oil is engineered to be as thin as straight 10 when cold but as thick as straight 30 when hot.
Viscosity matters because too-thin oil won't lubricate the engine well enough, and too-thick oil will reduce engine efficiency, make it difficult to start in cold weather, and increase the engine wear caused by cold starts. Oil gets thinner as it gets hotter; your owner's manual may recommend different weights depending on the temperature where you live. Also, your engine will have been built to expect a certain thickness of oil between its moving parts.
The other thing-on-the-label you might watch for is the quality rating. Any oil will have an American Petroleum Institute standard printed somewhere on the label; the oil you buy must meet or exceed the standard given in your owner's manual. You shouldn't have to worry about this unless your owner's manual goes out of the way to mention a minimum standard. Some manufacturers (such as Volkswagen) publish their own oil quality standards, in which case you'll want to do a little bit of Web searching to see if a particular variety of oil meets the automaker's standard.
Synthetic oil is generally considered better than conventional oil because it's more durable. Oil breaks down with time and use and becomes less effective as a lubricant. Synthetic holds up longer than conventional. Your engine ought to last longer if you use synthetic, but I don't know of any publicly-available hard evidence showing how much longer.
Provided you change your oil on schedule, however, you won't be doing anything wrong by buying conventional (non-synthetic) oil. As a middle ground, there are some "synthetic blends" available, which are a mix of conventional and synthetic oil. They cost less than "full synthetic" oil. (This is what I use.)
But the short answer is: do what it says in the owner's manual, and you'll be fine. You can read even more nitty-gritty details here if you're curious: