Learn Danish: Politikeren opfyldte ikke sit løfte. - The politician did not fulfil his promise.

Sit is a reflexive possessive pronoun. It is used because the promise belongs to the subject of the sentence, politikeren. If the sentence used hans (his) or hendes (her), it would mean the politician failed to fulfill someone else's promise.

Why is the pronoun spelled sit rather than sin?

The reflexive pronoun must agree in gender with the thing being owned, not the owner. Because løfte (promise) is a neuter noun (an et-word: et løfte), we use the neuter form sit. If the noun were common gender, like plan (a plan), you would use sin.

What would happen to the placement of ikke if this were a subordinate clause?

This sentence is a main clause, so it follows the standard V2 (verb-second) rule where the verb opfyldte comes before the adverb ikke. If it were a subordinate clause, ikke would jump in front of the verb. For example: Vi så, at politikeren ikke opfyldte sit løfte (We saw that the politician didn't fulfill his promise).

Is there a more everyday verb than opfylde to use with løfte?

While at opfylde et løfte (to fulfill a promise) is completely correct and slightly formal, it is very common in everyday Danish to use holde (to keep/hold). You will often hear Politikeren holdt ikke sit løfte (The politician didn't keep his promise).