Broach is a pointed tool for making holes: “The electrician used a broach to make a smooth hole in the wall.”
Keeping in mind the above meaning, there is another meaning of broach. It also means to begin to talk about something: “He wanted to broach a new topic for the discussion but the time ran out in the meeting.” In a way, this is an extension of the above physical meaning. Essentially, broach means to open, whether a hole or a discussion.
Brooch, on the other hand, is an ornamental pin used on one’s clothing: “Women generally wear beautiful brooches and do like to receive them as gifts.”