If you determine something, you figure it out or settle a question, like when, after checking paws and hands, you determine that it's not the dog that ate your candy stash, but your kid sister.
Determine comes from the Latin for "bring to a close," and when you determine something, you have closed all the questions you have about it and are quite sure you know the answer. If you become a detective, you'll have to determine who did the crime. After working through a mathematical problem, you determine the value of X. Determine can also mean influence. Your childhood spent in your grandmother's delightful kitchen might have determined your future career as a baker.