I could really use a cup of coffee

You're sleepy because you only got a few hours of sleep the night before. You're at work and really want a cup of coffee. You say this because there's no coffee available in your office.

I could really use a cup of coffee.

I could use (something)

You say "I could use ___" to indicate that you really want or need something, like a drink, but for some reason you haven't been able to get it yet. In the example at top, you say "I could really use a cup of coffee" because you're very sleepy and need some coffee, but there's no coffee nearby.

Here's another example - when you're struggling with something, and there's someone nearby who isn't helping you, you can say:

I could use a little help.

a cup of coffee

"Coffee" is usually an uncountable noun, so you count amounts of coffee using the cup:

I drink 2 or 3 cups of coffee in the morning.

You may sometimes hear people ask for "a coffee", but this is usually used when ordering coffee at a restaurant or cafe. In other situations, you should say "a cup of coffee".

really

"Really" means "very" but is better for casual conversation. "Very" is correct in written English, but can sound a little formal and old-fashioned in spoken conversation.

I thought it was a really original take on the story.

Also, "very" can only be used to modify an adjective: 

It's very small.

I'm very tired.

"Really" can be used for adjectives as well as verbs:

It's really small.

I'm really tired.

I really wanted to go

Do you really think so?