This question is deceptive. It sounds so simple, and yet it is very difficult to answer properly. There have been some wonderful answers, namely Bobby and Martin. There have also been some rather uninformed answers, but back to the question at hand. Both are simply car races where cars must conform to standards set by a governing body. They are excuses for people to strap themselves into something fast and try to beat the next person to the finish line. It's the same reason there are track and field races, ski races, bobsled, bike and speed skating races. It all boils down to competition. Both have their roots in the car races of the early 1900's. The European races of the 20's and 30's for F1 and the beach course races at Daytona Beach, Florida in the 30's. I'm not going to go into specifics here so that I can keep this answer to the point. You can find all the specifics you could want by going to the links provided in the "source" section below.
Now, let's divide this up for ease of comparison.
Formula One is open wheel racing, meaning simply that the cars' wheels are open, or not covered by the skin of the vehicle. The sport is basically technology driven. Manufacturers spend time and money in research and development to produce new parts construction materials etc. to make their cars faster than the next guy. The courses are generally road courses, a term meaning that the course consists of left and right turns as well as straights. As mentioned by others F1 races are sometimes run on courses made of blocked off city streets. They also race in the rain and use a different type of fuel. F1 races take place all over the world.
NASCAR is stock car racing, meaning that the cars are loosely based on 4 door sedans currently in production by the auto makers that compete in the series; Ford, Chevrolet, Dodge, and Toyota starting in 2007. The cars do use older technology, but they are meant to emulate the classic American hot rods. It's the idea of taking what you have and doing enough modifications to beat the other guy. Although, in NASCAR, those modifications must meet the standards and fit the rules. They compete on many different tracks, although most are considered oval. There are many configurations though; ovals, tri-ovals, quad-ovals, nearly rectangular, triangular, and even egg shaped like Darlington. They also race on two road courses each season. The oval type tracks differ in more than shape. They fit into three categories by length, "short tracks", "intermediate" and "super speedways". The tracks also have varying degrees of banking which, along with length, contribute to the attainable speeds. Two of the super speedways, Daytona and Talledega, use a restrictor plate because they also have high banking. The restrictor plate is a plate with regulation sized holes placed between the carburetor and intake manifold to restrict the flow of air and fuel, causing a reduction in power and speed. Unrestricted engines at these tracks can produce over 750 horsepower and reach speeds near 230 mph.
Many people say NASCAR is more exciting because of the crashes, while Formula 1 "is recognized as the highest class of auto racing in the world." Neither has any bearing on which series is better. That is based purely on opinion and you'll have to decide that for yourself. As you can see from the amount of information, I know more about NASCAR because I'm a fan. I even took a class in college called "The Evolution of Southern Motorsports" that went into all aspects of the sport. No matter what anyone else says, you should look at both series and make your own decision on which one you feel is better.