Proof-reading is the process of checking a proof draft copy of a typed or printed document in order to eliminate errors on the mechanical level, such as spelling and punctuation mistakes, double spacing and other typing errors. It is a painstaking process. Students should learn to proof-read carefully: it is an important skill in scholarship, although it is often seen as merely dull. It is certainly a skill in some ways of a low order - one would hope that students writing for Higher Education would be chiefly assessed on their performance of higher order skills, such as thinking, judgement, construction of an argument and research. But these do not flourish unless the text is well enough proof-read to be entirely unambiguous, clear and communicative. Badly proof-read writing cannot be guaranteed to be any of these. Proof-reading is rarely perfect, and the attempt to make it so may undermine efforts to improve the content; but it must always be done. See also consistency.
Since the earliest usage recorded in OED, in 1930, the process of proof-reading has sometimes been called 'proofing'. This is an informal usage. In academic writing, it is advisable (and never wrong) to use the fuller expression, however formal and 'stuffy' it sounds.
It may help non-native speakers of English to point out that like the verb 'to read', which gives proof-read its root, the present tense and past tense are homographs: they look the same, but are pronounced differently. See Read (homograph) or Read (irregular verb).