I see a lot of restaurants and grocery stores advertising kosher food. What qualifies a kosher dish?
There are quite a few rules involved. Short version:
That's the basics. The Wikipedia articles on kosher food and kashruth will explain further. Reform and Reconstructionist Jews may or may not follow some or all of the practices.
For meat, the answer is basically that the animal must have cloven hooves, and chew the cud. It must also be properly slaughtered, and the meat salted to remove the blood. There are all sorts of interesting edge cases to this, like giraffes, which are theoretically kosher animals, but no-one knows the way to slaughter them correctly. Also, fish (which must have scales and fins, I think) is not meat for this purpose, but poultry is.
You can't cook meat products and dairy products in the same set of pans, or serve them in the same dishes. There are all sorts of knock-on effects to this. For instance, some traditional cheeses are made with rennet, exracted from the stomach of a calf, which means that they've mixed "meat" and "milk" and are therefore unkosher.
Any leafy vegetable must be inspected to make sure there are no insects in it. grains and legumes need to be sorted to make sure that nothing inedible is accidentally included.
Any fruit or vegetable is unkosher if it was grown on Jewish land during the seventh year, when the land must rest. This rule doesn't apply for non-Jewish land.
A tenth of any food grown must be given to charity or left to rot. This is usually done by the growers, and if they're not Jewish then it doesn't usually matter.
That's all I can remember from my time in the Israeli Navy.
There's a lot of other details which are too numerous for this medium. There are also a lot of things to do with passover specifically, which you should probably ask about in a separate question.