Pairwork refers to organising a class so that students work together in pairs; eg in a class of 12, you'd have 6 pairs. Groupwork involves students working together in groups of 3, 4, 5, etc.
There are many advantages to pairwork and groupwork over work as a whole class. If students are working without their teacher, they learn to be more autonomous and can start to deduce or work things out for themselves. They are also more actively involved than when the lesson is entirely led by the teacher. An obvious advantage is when doing speaking work; if students are in groups or pairs, each individual will speak a lot more than during a whole class discussion.
While students are working in groups, the teacher can circulate and listen to what is happening. (S)he can take notes of good points that come up and any errors that will need focussing on later. It's important to do a feedback session after pair or groupwork to "finish it off" and clear up any outstanding questions the students may have.
Group and pairwork also provide the teacher with different ways of working with a mixed ability class. Students could do different exercises according to their level or they can be grouped so that stronger students can help weaker students. How a teacher groups students is important to consider; this can be used a way of separating noisy students and having students work with various people in the class.
Some of the disadvantages are that studets may get off-task and start discussing other matters unrelated to class work. They can also start talking in their mother tongue. You need to watch out for this when monitoring and be firm with students about speaking English.
Good luck with your course. I'm sure you'll learn lots of useful and practical things.