What exactly do one-to-one and two-to-one mean?

One-to-one (1:1) lessons are private lessons - just you and the teacher.  This means everything can be tailored to your needs and level.  You can "call the shots" but, at the same time, do listen to your teacher's advice on which areas you need to focus on.

Two-to-one (2:1) means two students taught together by the teacher.  The beauty of this arrangement is that it costs you a lot less each and you still get plenty of invididual attention.  Be aware, though, that you need to have both a similar level and similar aims and motivation.