When it's dropped as part of a way of speaking (which may eventually change the language as things drift over time), it's clipping. This is almost always losing one or more sounds at the start or end of the world - and it's how we get the word exam - from examination.
When it's done stylistically or for poetry to fit a metre, it's elision.
When it's a regular feature of English, or following the pattern of such a regular feature, and used to combine words, it's forming a contraction - like don't, can't, or Sam's (yes, the last one is a contraction, even when it's for the 'possessive', even though our language has lost the word that the -'s stands for2). Little fragments like -n't and -'s that are used in this way, at the end of a word, are called enclitics1.
1: Sam's might be "Sam is", such as "Sam's busy writing answers on ELL", or "Sam has", such as "Sam's been spending a lot of time on ELL", or the -'s can be a contraction of does or us in some situations. It can also be what's commonly called the possessive, as in "recently, Sam's usual method of procrastination is answering questions on ELL". The later type is referred to as the Saxon genitive as it forms the genitive (which covers possession, as well as a few other things, but we don't use that term as an everyday thing in English) and it comes from Old English (which owed rather a lot to the language brought over by the Saxons), and even then it wasn't a word - just a suffix, -es. In fact, there's some debate as to whether it was ever a separate word, rather than a pattern of inflection like -ing in modern English.
2: Some definitions I have read would exclude -n't from being any sort of clitic because the word it represents is still used separately, while -'s would meet all of the definitions I've seen when it's the Saxon genitive - but not when it's is.