What are the requirements for a dish to be kosher?
There are quite a few rules involved. Short version:


Animals must be slaughtered in a specific way, blood drained, and a rabbi must supervise
Dairy and meat (fish does not count as meat, generally) may not be served in the same dish. Depending on the level of observance, separate sets of dishes, cutlery, sinks, fridges, etc are maintained for dairy and for meat. Less Orthodox Jews will serve meats and dairy in the same meal, but not in the same preparation.
No shellfish; fish must have both fins and scales to be kosher
No pork
Only cloven-hoofed animals which also chew cud
A rabbi must supervise food preparation when prepared by non-Jews


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.