Watching movies and such where ships are brought along side of each other or chase each other down. How exactly is this accomplished if the wind is not in the ship’s favor? For example, the tv show Horatio Hornblower has a situation where a “fire ship” is sent into the British harbor at Gibraltar, how exactly is it propelled? If the wind decided to stop at that moment I guess the ship would just be dead in the water?
There's a very useful book on the subject by Sam Willis, "Fighting at Sea in the Eighteenth Century." It covers every aspect, from sighting to chase to close quarters to clean-up. Really good and easy to read, with diagrams for things like the situation you're envisaging.
Also useful is Harland's "Seamanship in the Age of Sail", though that doesn't cover the fighting aspect. Also has plenty of diagrams.
The fire ship would lose way if the wind dropped, yes. Fireships were horribly dangerous to use, which is why they fell out of favour. The crew would abandon ship at the vital moment and row off in a boat, but even that was hellishly dangerous. I've never come across a reference to fireships being towed. I think I got this info from Conway's History of the Ship: The Line of Battle, 1650 – 1840. Unfortunately my reference books are on the other side of the world so I can't be 100% sure of that!