A brooch is a piece of jewelry. It has a pin and a clasp so that it can be attached to clothing. The word is pronounced "broch," with a long o, and that leads to a common misspelling: broach.
A broach is a rod used for roasting meat or a sharp-pointed tool used for making a hole. The verb broach means to make a hole, as in a cask, or to open a discussion or bring up a subject.
Some dictionaries list broach as an alternate spelling for brooch, by the way, but most usage experts still separate the two spellings.
Both broach and brooch come from Middle English broche for a pin, peg or spit.