Please correct me if I'm wrong, but I'm not sure if this “a” should be there in the sentence (since coffee is uncountable). I put "Would you like some coffee?" and it was marked wrong :-( I suppose that the meaning of the sentence is "Would you like a cup of coffee?", but I usually don't see it written this way.
The "a" is completely fine (at least where I live, maybe it's not so common somewhere else). I would use "Would you like a coffee?" when, for example, someone visits my house to ask if they want me to get them a cup of coffee. Really, I would say that I hear/read "a coffee" somewhat more often than "a cup of coffee."
So the difference between "möchten" and "mögen" is the first is giving someone something that they might want?
Ok, i think it is totally pointless just to dropped us with this thing called "Preterite"!!! What the heck is preterite and its form Subjunctive II as it states in this site????? http://www.canoo.net/inflection/m%C3%B6gen:V:haben
Yes, "möchten" is, in fact, the preterite subjunctive of "mögen." But it's used so commonly that it's often treated as its own verb. Just know to translate it as "would like" or "want," and you'll be fine. You shouldn't meet any more preterites or subjunctives before lessons on these topics.
In any case, the preterite is the past tense. You'll often see it as the "simple past" in German, as German has two past tenses (simple past and compound past). Don't worry about it now--you'll learn about it later.
The subjunctive shows hypothetical ideas or wishes (e.g., "If I lived in Germany [but I don't, so this is just hypothetical], I would know German."--here, we would put "lived" and "would know" in the subjunctive).
Again, don't worry about these until later; just know that "möchten" is "would like."