Contractions occur when two words are combined into one, sometimes with a distinctly different spelling. In English, contractions like "won't" are optional and indicate informality. In French, however, contractions are required, regardless of the register you're speaking or writing in.
The letter e has several different pronunciations in French. The explanation and equivalents provided here are based on the closest sounds that exist in American English, which are sometimes not very close at all.
In many words the letter e is potentially silent, a characteristic which has three French names: e caduc, e instable, and e muet. Though e muet is the most common term, e instable is the most accurate.
An elision is a type of contraction that occurs when two words are combined: one or more letters are dropped and replaced with an apostrophe. In French, written elisions are required, regardless of the register you're speaking or writing in.
Whenever a word ending in a consonant sound is followed by a vowel or h muet, that consonant sound is transferred onto the next word. This is called enchaînement and it's one of the aspects of French pronunciation that sometimes makes it difficult to determine where one word ends and the next begins.