(2-part post):
The majority of solicitors (and barristers) start off by studying law at university, and then go on for further study to enter their chosen profession. However, plenty of people study law and don't go into either profession. (They might,for example, go into business administration). LLB after someone's name simply means that they've obtained a 'first' (= 'bachelor's') degree in law. It's not specifically a qualification related to being a solicitor.
All bachelor's degrees (in whatever subject) are 'taught' degrees. (i.e. the students attend lectures and tutorials). A student who does well in their degree examinations (typically obtaining a 1st class honours degree or a 2.1 degree) might be accepted to study for a further (master's) degree. This is usually a one year course and is most commonly, once again, a 'taught' degree. If a law student successfully undertakes such a degree course, he'll become a Master of Law and be entitled to put LLM after his/her name. (Note, however, that this would normally happen before the person had even started on the professional studies to become a solicitor or barrister. It doesn't mean he's a better solicitor!).