To litigate is to engage in some sort of legal proceedings. Litigating mostly takes place in court.
If you're litigating, you'd better have a lawyer: you're involved in some sort of legal situation. When you sue someone, that's litigating. The person sued is also forced to litigate. People accused of a crime have to litigate when they're on trial. There are many types of litigation. Lawyers usually specialize in one type of litigating, such as criminal law or corporate law. If you're ever on a jury, you'll get a closer look at how litigating works.