It can be hard to apply these words correctly: they sound exactly the same, and their spellings aren't easily connected to their meanings. One of the pair, however, is quite limited in scope, and focusing on when to apply that one can be key.
When the subject is slowing or stopping movement, the word to use is brake. Brake is both a noun, as in "put on the brakes" and "took my foot off the brake," and a verb, as in "brake at the stop sign" and "I brake for moose." As a noun, it's also used before other nouns: "brake fluid," "brake pedal." As in these examples, the word is used in contexts relating to cars, bicycles, and other vehicles. It's also used figuratively, however, to talk about the slowing or stopping of activity, as in "putting the brakes on spending."
Break also functions as both a noun and a verb, and it's the word you want in all other contexts, such as when the topic is something separating into parts or pieces ("the plate will break if it falls," "break a leg," "a bad break"), being damaged to the point of no longer working ("break a watch"), failing to do what is promised ("break a promise"), or referring to a time during which activity stops ("take a break").
If you have difficulty keeping these straight and are inclined to think in pictures, you might want to imagine a foot nestled in the top of the k in brake, pressing that top line down onto the e, which isn't saying a thing, because the k has put the brakes on.
