To apologize: s’excuser
The verb is not reflexive in French but reciprocal as you can’t “excuse” yourself but someone can “excuse” you or you can “excuse” someone else. Even if French people use the expression “je m’excuse” it is not correct French.
You want to “excuse” someone:
Je l’excuse (he or she)
Je vous excuse (formal or plural)
Je les excuse (them)
Somebody else wants to “excuse” someone:
Tu nous excuses
Tu les excuses
But not: tu t’excuses! (however it is common to hear that in France, even in newspapers)
You want to “excuse” yourself:
Excuse-moi (you ask to someone to “excuse” you)
We also use these expression to say: I am sorry or I beg you pardon.
But not: je m’excuse (however it is common to hear that in France)