A contraction is a short form of a word or group of words, written with an apostrophe instead of the full form. Can't is short for cannot, and I'm is short for I am. The word also has unrelated meanings, like a muscle tightening.
In a message, someone writes I couldn't find my keys this morning. Here couldn't is the short form of could not: the word not loses its o, and the apostrophe marks the gap. The same works in a question: Couldn't you leave on Saturday instead?
It does not only happen with not. In She doesn't like coffee, doesn't is the short form of does not, with the apostrophe sitting where the missing o used to be.
The apostrophe is the tell. It sits exactly where letters are dropped, so in they're the mark replaces the missing a of they are, and in isn't it replaces the o of is not. If you can expand the words and the apostrophe marks the gap, you are looking at a contraction.