A worked example
With the keyword LEMON, ATTACKATDAWN becomes LXFOPVEFRNHR. Each letter of the key is a Caesar shift, A for 0 up to Z for 25: the first letter moves by L (11), A to L; the second by E (4), T to X; the third by M (12), T to F; and so on, starting the keyword again when it runs out. Decoding runs the same shifts backwards.
The keyword and the spacing
Only the letters of the keyword count: spaces, digits and capitals in it are ignored, so Lemon, LEMON and le mon are the same key. In the message the key moves on only at letters, so spaces and punctuation stay where they are and do not use up the keyword. A key of A changes nothing, and a one-letter key is a plain Caesar shift.
Questions
Can it decode without the keyword?
No, it cannot find the key for you. In puzzles the keyword is usually hidden nearby, such as a name, a place or the theme of the game, so try those first. Long messages can be broken by counting letters at each position of the key, but that is work for pencil and paper or a dedicated solver.
Why was it called unbreakable?
Each letter can be shifted by a different amount, so simple letter counting does not crack it, and it was known as le chiffre indéchiffrable for about three centuries, until Charles Babbage and Friedrich Kasiski showed in the 1800s how to find the length of the key.