Learn Hardcore German: Ich klebe einen kleinen Merkzettel an den Spiegel, damit ich den Termin nicht vergesse. - I stick a little reminder note on the mirror so that I don’t forget the appointment.

An is a so‑called “two‑way preposition” (Wechselpräposition) in German. With these prepositions, the case depends on whether you are talking about:

Location (no movement to a new place) → dative
Direction / movement toward a place → accusative

In Ich klebe einen kleinen Merkzettel an den Spiegel, something (the note) is being moved to the mirror and ends up on it. This is a change of location, so you use the accusative:

an den Spiegel (accusative, masculine singular)

If you were just describing where something already is, you would use the dative:

Der Merkzettel hängt an dem Spiegel / am Spiegel.
“The note is hanging on the mirror.”

So:

Movement to the mirror → an den Spiegel (accusative)
Already at/on the mirror → an dem Spiegel / am Spiegel (dative)

Could I also say auf den Spiegel instead of an den Spiegel?

You can, but it changes the mental picture a bit.

an den Spiegel literally means “to the mirror,” more like “onto the mirror / on the mirror’s surface,” but idiomatically it’s used the same way as an die Wand (on the wall). It suggests putting the note on or at the mirror, as an object mounted on the wall.
auf den Spiegel literally means “onto the mirror (on top of it).”
Native speakers would usually reserve auf for horizontal surfaces (e.g. auf den Tisch, on the table). Using auf den Spiegel can sound like you’re putting something directly onto the glass surface as if it were a flat surface you place things on, which is less typical.

In everyday speech, for a note stuck to a wall mirror, an den Spiegel is the most natural choice. You’ll very often hear:

Ich klebe den Zettel an den Spiegel.
Ich hänge etwas an die Wand.

Why do we say einen kleinen Merkzettel and not ein kleiner Merkzettel?

Because einen kleinen Merkzettel is in the accusative case, and Merkzettel is masculine.

Function in the sentence
Einen kleinen Merkzettel is the direct object of klebe (I stick/glue something). Direct objects in German take the accusative.
Article and adjective endings (masculine, accusative singular)

Nominative (subject):
- ein kleiner Merkzettel – “a small note” (as subject)
  - ein (masc. nom.)
  - kleiner (adj. ending ‑er)
Accusative (direct object):
- einen kleinen Merkzettel – “a small note” (as object)
  - einen (masc. acc.)
  - kleinen (adj. ending ‑en)

Since in the sentence the note is the thing being glued (direct object), you must use the accusative form:

Ich klebe einen kleinen Merkzettel an den Spiegel.

What exactly is a Merkzettel, and what gender is it?

Merkzettel is:

a noun (hence the capital letter in German)
masculine: der Merkzettel
plural: die Merkzettel (singular and plural look the same in writing)

Meaning:
A Merkzettel is a note to remind you of something – a “reminder note,” “memo,” or “little list of things to remember.”

Related word: Zettel (also masculine: der Zettel). Zettel is a more general word for a “piece of paper / slip / note.” A Merkzettel is specifically a note with something you want to remember (an appointment, tasks, shopping items, etc.).

Examples:

Ich schreibe mir einen Merkzettel. – I write myself a reminder note.
Auf dem Merkzettel steht der Termin. – The appointment is written on the reminder note.

Why is the verb at the end in damit ich den Termin nicht vergesse?

Damit introduces a subordinate clause in German. In subordinate clauses, the conjugated verb goes to the end of the clause.

General pattern:

Main clause: verb is in second position
- Ich klebe einen kleinen Merkzettel an den Spiegel, ...
Subordinate clause (introduced by damit, weil, dass, wenn, etc.):
- ..., damit ich den Termin nicht vergesse.
- ..., weil ich den Termin nicht vergessen will.
- ..., dass ich den Termin nicht vergesse.

So:

Main clause: Ich klebe ...
Subordinate clause: damit ich den Termin nicht vergesse

This “verb at the end” rule is standard for subordinate clauses in German and is one of the key word‑order differences compared to English.

What does damit express here, and how is it different from um ... zu?

Here damit introduces a purpose clause: it explains why you stick the note on the mirror – in order not to forget the appointment.

damit = “so that / in order that”

You could rephrase:

Ich klebe einen kleinen Merkzettel an den Spiegel, damit ich den Termin nicht vergesse.
“I stick a small reminder note on the mirror so that I don’t forget the appointment.”

Difference from um ... zu:

Subject of the action

damit can be used even if the subjects are different in main and subordinate clause:
- Ich klebe einen Merkzettel hin, damit du den Termin nicht vergisst.
  (I ... so that you don’t forget.)
um ... zu can only be used when the subject is the same in both parts:
- Ich klebe einen Merkzettel hin, um den Termin nicht zu vergessen.
  (I do X in order not to forget the appointment – same “I”.)

Structure

damit
- full clause with a finite verb at the end:
- ..., damit ich den Termin nicht vergesse.
um ... zu
- infinitive:
- ..., um den Termin nicht zu vergessen.

In your sentence both are grammatically possible and natural:

..., damit ich den Termin nicht vergesse.
..., um den Termin nicht zu vergessen.

Using damit puts a little more focus on the whole clause as a purpose; um ... zu sounds a bit more compact.

Could I replace damit with so dass or also here?

Not without changing the nuance or the grammar.

so dass / sodass
This often expresses a result rather than an intended purpose.

Ich klebe einen Merkzettel an den Spiegel, sodass ich den Termin nicht vergesse.
This can sound more like: “I stick a note on the mirror, with the result that I don’t forget the appointment.”
It’s not wrong, but damit is more clearly about my intention.

also
Also cannot introduce a subordinate clause like damit. It’s a sentence adverb used in a main clause:

Ich klebe einen Merkzettel an den Spiegel, also vergesse ich den Termin nicht.
Here you now have two main clauses:
“I stick a note on the mirror, so I won’t forget the appointment.”

So:

To express purpose/intention → damit (or um ... zu).
To express a result → sodass / sodass.
To join two main clauses with a consequence → also.

Why is it vergesse (present tense) and not something like vergessen werde to talk about the future?

German usually uses the present tense to talk about the near future, especially when context makes it clear that you mean the future.

..., damit ich den Termin nicht vergesse.
Literally: “... so that I do not forget the appointment.”
But it means: “... so that I won’t forget the appointment (in the future).”

Using werden here would sound unusual or overly heavy:

..., damit ich den Termin nicht vergessen werde. – grammatically possible, but rarely said in everyday language here.

So in German:

Present tense + time context often = future meaning.
Ich gehe morgen zum Arzt. – “I’m going to the doctor tomorrow.”
Ich rufe dich später an. – “I’ll call you later.”

Likewise:

..., damit ich den Termin nicht vergesse. = “... so that I don’t / won’t forget the appointment.”

Why is nicht placed before vergesse in damit ich den Termin nicht vergesse?

In German, nicht usually comes:

before the verb (or the verb cluster) it negates,
but after most objects and specific details.

In a subordinate clause with the verb at the end, the typical order is:

Subject – objects / adverbials – nicht – verb (at the end)

So:

damit ich den Termin nicht vergesse
- ich (subject)
- den Termin (object)
- nicht (negation)
- vergesse (verb)

You could say damit ich nicht den Termin vergesse, but that sounds like you are contrasting this appointment with another thing:

..., damit ich nicht den Termin vergesse, sondern etwas anderes.
(“... so that it’s not the appointment I forget, but something else.”)

Without that contrast, the neutral word order is:

damit ich den Termin nicht vergesse.

Could I say Ich klebe mir einen kleinen Merkzettel an den Spiegel? What is the difference?

Yes, you can, and it’s very natural.

Ich klebe einen kleinen Merkzettel an den Spiegel.
Neutral: “I stick a small reminder note on the mirror.”
Ich klebe mir einen kleinen Merkzettel an den Spiegel.
More specifically: “I stick a small reminder note on the mirror for myself.”

Here mir is a dative reflexive pronoun, showing that the action is for your own benefit. This is similar to English saying:

“I’m making myself a note.”
“I’m putting a note on the mirror for myself.”

Both versions are correct; adding mir simply emphasizes that you are doing it for your own use / benefit.