Basic grammar question: what are contractions in writing and when should I use them?

[RobShein]

I feel silly asking this, but I'm a freshman and I keep hearing about contractions in my English class. I know they're things like 'don't' and 'can't,' but I need a clearer definition of what are contractions in writing. My textbook says they're shortened forms of words, but that seems too simple. Are there rules about forming them? My teacher mentioned something about apostrophes replacing missing letters, but I'm still confused about where the apostrophe goes. Like, why is 'won't' the contraction for 'will not'? That doesn't follow the pattern at all. And what about 'it's' versus 'its'? I mix those up constantly. If anyone has a simple explanation or a cheat sheet for common contractions, I'd really appreciate it. I want to use them correctly, not just guess.