Since I hold a couple of LL.B’s and an LL.M., I thought I’d chirp in.
The historical sequence of degrees in England was LL.B., LL.M., and the LL.D. - Bachelor, Master and Doctor of Laws, respectively. However, that logical sequence has been disrupted.
The LL.D. long ago became an honourary degree. For real work, it’s been replaced by either the JSD or DCL, as King Rat mentioned.
In the United States, the LL.B. was replaced with the J.D. (“Juris Doctor”) when the powers that be decided to make it a graduate degree (i.e. - you could only apply if you had a bachelor’s degree of some sort, like a B.A., B.Sc., etc.)
However, in Canada and England, and I believe other Commonwealth countries, the LL.B. is still technically not a graduate degree. You can, in theory at least, be admitted to the LL.B. program without a prior degree. For example, in the common law provinces of Canada, you need at least two years of university work, but technically you don’t need a degree (as a matter of practice, few students get in who don’t have a previous degree, but it’s not required). In Québec, you can get into the law program out of CEGEP, and in England straight out of the equivalent of high school.
That leaves the LL.M. As stated, it can be an opportunity to specialise in a particular area of law, or it can be a stepping-stone to the doctorate program - the University decides to take you on for one year and see how you do, before admitting you to the doctoral program.