ACORDARSE means ¨remember something which we may forget or have forgotten¨, so I should say:
No me acuerdo. I don´t remember.
Acuérdate de llamarlo. Remember to call him.
(Notice the use of to+infinitive when we refer to something that needs to be done).
Siempre me acordaré de ti. ¨I will always remember you.¨
Acordarse is a pronominal verb (reflexive verb), we need to use the pronouns ¨me, te, se, nos, os¨ when using it.
RECORDAR means ¨recall¨ with the idea of bringing back to your mind something that happened in the past:
Recuerdo cuando fuimos a visitarlo. I remember visiting him.
(Notice the use of -ing after remember when we refer to something that has already happened).
Siempre lo recordaré. I will always remember/recall it.
It can also mean ¨remind¨:
Recuérdame que le llame mañana. Remind me to call him tomorrow.
Me recuerda a mi padre. He reminds me of my father.