The term al dente comes from an Italian phrase which translates as "to the tooth." When cooked al dente, pasta should be tender but still firm to the bite. It shouldn't be mushy, as mushy pasta is the bane of Italian chefs.
Some chefs suggest that when you bite into a piece of pasta which is cooked al dente, you should see a tiny white dot in the center of the pasta. Others say that al dente is when the dot is no longer there. The dot represents the center of the pasta being slightly undercooked. Either way, al dente pasta should have a bite to it.
To test for al dente you can start biting into the pasta a minute or two before the package instructions indicate it should be done. When you bite into it and your teeth feel some resistance, but the pasta is still tender, you've reached al dente.
Note that al dente will feel different with fresh pasta than it will with dried pasta, and fresh pasta only needs to be cooked for a very short time. Still, even with fresh, the cooked pasta should have some bite to it. In fact, making sure you don't overcook it is even more paramount with fresh pasta, as you will have much less margin for error.