Archaic just means 'old', mostly in the sense of 'old-fashioned' or 'out-of-date'. A contraction is simply a 'shortening' - in the study of language, usually a shortening of words.
Some words in modern English are the result of shortening older, longer, words. The word 'car' was, in its modern sense, a motor-car; a 'bike' of course is properly a bicycle. This is a normal process of linguistic change. However, some pedants like to show awareness that it has happened (and, by so doing, indicate that they do not approve). It seems to AWE to be natural that when a word is being changed, some people will object to it. But it is unfortunate to continue the practice long after the new abbreviation has become accepted in society generally. Modern students should avoid it.
- (In the nineteenth century, contractions were marked more often than nowadays. For example, you will see the shortening of 'shall not' written as sha'n't in many novels printed in the Victorian period.)