Learn German: Ich vergesse meinen Termin nicht. - I am not forgetting my appointment.

Termin is a masculine noun (der Termin). In this sentence, the appointment is the direct object (the thing being forgotten), so it must be in the accusative case. For masculine nouns in the accusative, the possessive word mein takes an -en ending, becoming meinen.

If I wanted to say "I am not forgetting YOUR appointment", how would I change this?

You would use the accusative form of "your". If you are talking informally to a friend, you would use deinen (Ich vergesse deinen Termin nicht.). If you are speaking formally to someone, you would use Ihren (Ich vergesse Ihren Termin nicht.).

Why does nicht go at the very end of the sentence?

When you negate an entire action or idea—in this case, the act of forgetting your appointment—nicht usually gets pushed to the end of the clause. While English puts "not" right inside the verb phrase ("am not forgetting"), German places the direct object (meinen Termin) before the nicht.

The English translation says "am not forgetting", but doesn't vergesse mean "forget"?

It means both! German doesn't have a separate "-ing" continuous tense like English does. A present tense verb like vergesse covers both "I forget" and "I am forgetting". You just rely on the context to choose the most natural English translation.