"Fix" is a very broad term meaning to correct almost anything that is wrong.
For example, "There's one small error in that sentence. You need to fix it." The word "fix" can be used here. "Repair" and "mend" do not.
"Repair" is often used with machinery or buildings. "I have to take the car to the mechanic and have it repair." "The roof is leaking, I have to have it repaired."
"Mend" is often used with respect to clothing, paper, or material, including wood, or even a broken bone, that has come apart and needs to be joined together again.
"I see that your shirt got torn. Did it get caught on barbed wire again? It doesn't look bad, give me a minute with the sewing machine and I can mend that rip." We could also say "fix that rip" or "repair that rip," but "mend" is more natural.
"I had a bad fight with my friend, and we haven't spoken in several days. I'm going to talk to him and apologize, and try to mend our friendship."
1. Repair, to me, means something has major damage and is going to take a while to fix or at least a bit of time.
ex. The bridge crossing that lake is undergoing some repairs.
2. Fix means the same thing as "repair" but, to me, has less urgency in its meaning.
ex. I need to fix the tv, I don't know what is wrong but the buttons on the remote don't seem to register with the tv.
3. Mend also means fix/repair but its usually used on things like clothes or fabrics.
ex. I'm going to take these pants to the tailor so they can be mended.
Just as a side note: these are just the nuances that I have derived from these words nothing that is official.