A couple is describing "2" of a certain thing. Sometimes it can be more than 2, but a small amount.
I want a "couple" of donuts. That "couple" is cute (Boyfriend/Girlfriend).
A pair is describing something that comes in "2". Some words use "pair" even if it's one item though.
I will a buy a "pair" of socks (socks come in 2, usually we don't buy 1 sock).
That's a nice "pair" of shoes!
Ex. of using "pair" with one item:
Bring a "pair" of shorts to the swimming pool.