The word critical often gives pause to new students, and sometimes more experienced ones. The idea behind the phrases 'Examine critically ... ', 'Critically assess ... ', 'Give a critical account of ... ' and so on - very common in assignment titles and exam questions - is fundamental to a University education. In science, there is a different meaning of critical; see critical (physical sciences).
The word family critic, criticise, critical, criticism and critique share a derivation from the ancient Greek word κριτής‚ (critēs), which means 'judge'. In academic English, words like 'critic' and 'critical' involve the idea of judgement. AWE's advice is never to forget the idea of a judge in a court: you must come to a considered judgement, having weighed up the evidence on both sides.
The difficulty is that in colloquial English, the idea of 'criticism' is limited to negative judgements. 'Oh, you're always criticising me' doesn't convey an idea of academic objectivity and impartial judgement. Remember that 'This is the best account of ... ', or 'This is an extremely well-written essay' are criticisms in the academic sense. This more positive use may be rarer, even in the University, than the negative; but it is part of the idea of critical judgement that we would like all students to develop.