Here is how the words are used in my part of the country. The difference is not so much in meaning as in usage. The usage in other parts of the country maty vary. In general, fix is used when we want to put sth back into its original condition or restore a situation to its former state. Fix emphasizes the existence of a problem. The door is stuck. I have to fix it. Maria is mad at me. I have to fix the problem. I can’t drive my car until I fix it. Repair is often used when we refer to the actual components of the problem. Repair emphasizes damage to certain things. I can’t drive my car until I repair the starter motor. He bought plumbing supplies to repair the shower faucet. After the hurricane, the repairs to my house cost more than five thousand dollars. I got mad at Maria and used impolite language. Now I have to repair the damage done to our relationship. Mend is often used when referring to damage to clothes or fabric in general. I ripped my pants. I have to mend them before I go out. I have to mend the hole in my sock. The tear in the curtains has to be mended. Mend is also used in reference to certain objects, I have to mend the window screens. The mosquitoes are coming in. (I am of an older generation and perhaps younger Americans would not use mend in this way, but for my generation it’s not uncommon…..at least in the South.)

Fix - is the most commonly used, because it's short and has a very wide meaning range - not only literally for repairing something, but also for attaching or mounting, sceuring. You can say "I'll fix that" for whatever is to be done. Repair and mend don't these alternative meanings. Repair - a quite neutral and precise way to say you fixed something broken :-) Mend - is a quite old-fashioned verb, but means the same as "repair".

Here are the dictionary definitions: mend = a verb Meanings: to repair something that is broken or damaged: Could you mend this hole in my shirt? I've left my watch at the jeweller's to be mended. The plumber came to mend the burst pipe. The country's president is seeking to mend relations with the United States. repair= a verb Meanings 1 to put something that is damaged, broken or not working correctly, back into good condition or make it work again: to repair (the surface of) the road to repair a roof after a storm The garage said the car was so old it wasn't worth repairing. I really must get my bike repaired this weekend. fix = a verb 1. to repair something: They couldn't fix my old computer, so I bought a new one