BSc is a standard 3 year science degree course. MSc is a 1 or 2 year post graduate science course which is similar to the degree (ie mostly going to lectures and doing exams). You have to have a degree to do a post graduate course. MPhil is another 2 (or so) year post graduate course but is based on research: you research a new subject and write a thesis on it. It can be any subject. Similarly, a PhD is like an MPhil but this lasts for 3 (or so) years and you get to call yourself a doctor. A MEng is a specific MSc course that you do in engineering. In other words it is a post graduate course in engineering that has no element of research.
In the US, we don't use the Sc, just the S..... Thus you graduate with a B.S. or an M.S. .... Lots of jokes about what the B.S. stands for.... Also there used to be jokes about girls going to college to get their M.R.S. (Mrs.).... And also jokes about grad students' working wives who earned their PHT (Putting Hubby Through).