A bachelor's degree such as a BA, BSc, or BEng is a three year undergraduate course. They typically consist of regular lectures, coursework, and exams. Science and engineering courses will also include labs and other practical sessions. These are usually undertaken after completing A-Levels or equivalent.
Master's degrees are a higher level of study than bachelor's degrees, and involve studying a topic in more depth. Most technical roles and PhDs require you to have a master's degree.
If you already have a bachelor's degrees then you can study a specialist topic for one or two years to obtain a postgraduate master's such as an MSc or MA. This can either be a taught course with lectures, or a research course involving independent study.
If you don't have a degree you can study for an integrated master's. This is like a bachelor's degree but lasts for four years and leads to an MEng, MPhys, or MSci*. Some universities only offer integrated masters for certain courses, particularly engineering.
Although on paper they are the same level qualification, an MSc allows you to specialise more in your field whereas a combined masters is a continuation of your undergraduate degree but to a higher level. A research master's is particularly good preparation for a PhD and is required by some universities. These are often funded in the same ways as PhDs. If you're worried about being an older student, don't be - MScs tend to be a popular option after a few years of work to open doors for the field you want to go into.
*These are the most common abbreviations for master's courses, but there are many different ones and not all variants are listed here.
For undergraduate bachelor's and master's courses, government grants and loans are avialable through Student Finance regardless of how long your course is.
For postgraduate master's courses, there are a number of grants and scholarships available as well as the option of taking out a Career Development Loan (different to Student Finance as you have to repay it whether you are earning or not).
If you do a course that includes study abroad you can also apply for an Erasmus grant.