What exactly is a Masters of Laws (LL.M)?

As with any Master's, it's usually a degree where you study more about a particular area of law, having already obtained a law degree. There are a few countries (Denmark, Finland, Sweden, the Netherlands, Norway, Cyprus, Spain, Italy and Switzerland) where it is the basic degree you need to be a lawyer, but mostly it's a specialist extra once you already have your basic law degree, like you would do an MA after your BA. Often it requires writing a thesis, but it may not.

I see from previous questions you're American. In the USA, you need a JD (and pass the professional exams) to become a lawyer, and with a JD you could do an LLM in a more specialised area. Tax law is quite popular. An LLM is completely optional - to be a lawyer, it's the JD you need, plus whatever your state's bar association requires to qualify to practise.

In other English-speaking countries (which I know better because I'm British!), it's normal to do your bachelor's degree in law, and almost always this is an LLB. (At Oxford and Cambridge it's a BA, but then all bachelor's degrees there are BA, even science.) Not like the US, where you do a more general degree first, then go to law school for your JD. So LLM is a natural progression from LLB, like MA from BA or MSc from BSc. The US seems odd to me in that you go from a bachelor's degree to a Doctor of Law, then kind of backwards to Master's if you want to study a particular part of law further!

It varies a bit by country.

In the U.S., most who practice law first obtain their JD (Juris Doctorate). Some people who want to get more knowledge in an area will pursue an LL.M afterward, where they essentially go do a year of specialized research resulting in a thesis paper, much like someone might also do a Master's after undergrad.

LL.M degrees also exist in Europe, where they can even be the degree required to practice law.

In the U.S., you'll sometimes see almost exclusively foreign students in an LL.M program at a school, where they'll spend a year learning the U.S. legal system and taking some of the 1st and 2nd year law student courses. But since they already have a law degree from their home country, that can sometimes be enough to get to practice in some aspects of the U.S. system, or at least better advise law firms in their home countries how to handle things in the U.S.

The abbreviation you used is for a "Masters of Legal Letters" and not Masters of Laws.. They are slightly different Masters degrees with several different subjects. A Masters degree a higher degree than a Bachelors and is often the second degree a law student aims for. It is earned by studying several subjects and completing a 25,000 word thesis or by completing a 40,000 word research thesis and completed over rwo years (full time).

It is a one year graduate program in one, you pick, specific part of the law.
You do not need a law degree nor to have passed the bar to get the degree. It really is basically a certificate, not a real masters degree in anything.

A type of law degree. The other is Juris doctorate.

Overkill.