What is the grammatical aspect in Greek for?

The grammatical aspect indicates whether an action is/was completed or not. 

The perfective aspect (as in aorist, aorist infinitives) shows that the action was completed, often that it was a successful or one-off event, while imperfective aspect (imperfect, present infinitive,) expresses that the action was incomplete, it’s not important whether it was completed or not, it extended over a longer time or was unsuccessful.

It’s easiest to illustrate it with simple examples.  

Τήν μητέρα μουἔπεισα (aorist)  = I persuaded my mother. (My mother is therefore successfully, completely persuaded)

 

Τήν μητέρα μου ἔπειθον (imperfect) = I persuaded my mother for some time (I tried to persuade my mother, but I failed. My mother is therefore not persuaded)

 

Ἐθέλω τήν Ἐλλνηκήν γλῶσσαν μάθειν (perfect infinitive) = I want to learn Greek  (my aim is to complete learning that is to be a proficient user of Greek)  

 

Ἐθέλω τήν Ἐλλνηκήν γλῶσσαν μανθάνειν  (present infinitive) = I want to learn Greek  ( my aim is the learning itself, I want to start learning the language, it’s not important if I ever become proficient)