“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, but there's no coffee available in your office. You say:

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.

really

"Really" is similar in meaning to "very". But while "very" can only be used to modify an adjective ("very small", "very tired", etc.), "really can be used for adjectives ("really small", "really tired") as well as verbs ("I really wanted to go", "Do you really think so?")

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".