Something that's incongruent doesn't fit. If you saw your teacher speeding around town in a red sports car, blasting music, this would be incongruent with your image of her as very reserved and studious.
Use incongruent to describe something that's out of place or incompatible, like the orange house among all the white ones, or a primly dressed little girl who tells dirty jokes at the second grade talent show. In math, two numbers are incongruent when, after being divided by the same number, their remainders are different. The Latin roots of incongruent are in, or "not," and congruentem, "suitable or agreeing."