In math and science, a coefficient is a constant term related to the properties of a product. In the equation that measures friction, for example, the number that always stays the same is the coefficient.
In plain English, coefficient means "joining together to produce a result." Sometimes people use the word to talk about social phenomena, like the coefficient factors of vanity and self-loathing in a celebrity's alcoholic demise. But mostly you'll encounter it in math and science. In algebra, the coefficient is the number that you multiply a variable by, like the 4 in 4x=y. In chemistry, when you see a number in front of a chemical like 2H2o, you're looking at the coefficient.
