"I am drinking your coffee!"
Translation:Ich trinke deinen Kaffee!
Kaffee is a masculine word (like most drinks, except beer). Remember der Kaffee and ein Kaffee.
In this sentence there is a subject (ich), a verb (trinke) and an object (Kaffee). Like most verbs, trinken puts the object into accusative case.
In accusative case der Kaffee becomes den Kaffee, and ein Kaffee becomes einen Kaffee. If that's not clear to you, revise the lesson in the previous link.
The possessive determiners (all the words like dein-, sein-, ihr-, mein-, unser-, etc. that are used to show possession with a noun after them) need an 'ending' that matches the form of ein/eine/einen/etc. that would be needed in the same position (matching the gender and case of the noun). I find this easiest to remember when I think that ein sounds like dein.
Because we would need einen Kaffee here, if we instead use dein- the ending needs to match einen, i.e. we need to use deinen.
Does that explain it clearly? Here is more information and a practice exercise.