'Mix' means 'combine' : to mix, all you need to do is put different things together.
A bag of mixed nuts might contains peanuts, almonds and hazelnuts. A mixed class might have beginner and intermediate students in it. Sometimes you have to stir <em>in order to </em>mix together well (cake ingredients, for example), but sometimes you don't. Cocktails, for example - the best cocktails are shaken rather than stirred ( as all James Bond fans know).
'Stir' is a movement: for example, you put a spoon in a cup of liquid and move the spoon in circular movements so that the liquid moves.
It's possible to stir without mixing anything together: for example, if you have a cup of containing just coffee, you can stir it to to cool it down.