LLM stands for Master of Laws, an advanced legal degree. In the
American judicial system, the LLM does not entitle recipients to take the
bar exam or practice law; those privileges require a Juris Doctor (JD)
degree. Oddly enough, the LLM ranks higher than the JD, even though the
former is a master's and the latter is a doctorate. (According to a
cynical lawyer friend, the LLM was originally the highest law degree
until a bunch of lawyers decided that they needed a doctorate too, just
like the physicians and the professors out
there.)
LLM programs generally allow students to specialize in a particular
area of law, such as taxation, international law, business, health
policy, and so on. They usually require one school year; some programs
require a thesis. Also, Sekicho tells me that foreign lawyers often get an LLM to practice in the US.