The communicative approach to language teaching developed as a way of introducing ‘real' communication and language into the classroom.
It was a reaction to exercises of the type: reading aloud, listening and repeating, translation and the sort of speaking activities that involved everyone in the class looking at a picture with the teacher asking questions about it and the students answering. You can see that there was no real reason to communicate here as everybody knew the answer.
Communicative activies cover reading, writing, listening and speaking. However, many people associate the term with communicative speaking tasks. If you are following a coursebook, you are probably using the approach without realising it.
A communicative reading or listening activity might involve students reading or listening to directions of how to get through a town and following the directions on a map. Writing could involve an exchange of letters in the classroom where students ask and answer each other’s questions.
Role play is a good example of a communicative speaking activity. Students are given a role to play (which could include information about their role's personality, opinions, etc) and a set topic to discuss. They have time to prepare their role and what they want to say before starting the discussion.
Questionnaires are also used. This could involve students doing a survey of class members’ eating habits, preferred forms of transport, recent holidays, personal possessions or any other topic.
Information gaps involve students having different but complementary information. They need to ask questions in order to complete a task. For example, a table with information on 6 people including addresses, phone numbers, place of work, date of birth etc. Each student has some information to complete the table but needs to ask other students questions to obtain all the details.
You could also give one student a picture which (s)he describes to his/her partner. The partner draws the picture.
‘Spot the difference’ activities can be used where there are 2 pictures identical except for 10 small differences. Each student has a picture that only (s)he can see. The differences are found orally by describing what can be seen and asking for information about the partner's picture.
You can see that in all the above, there is a real need for communication as information needs to be shared. When there is real speaking, real listening is also required.
This is a very brief overview of communicative language learning, a subject that has been written about at length. Due to limited space, I can't go into more detail about the activities. You could contact me again for further information.