студент(ка) is an university student, étudiant(e).
It is rather your (formal) status, like "boy" or "programmer". Also identity to some extent: there are lifestyles characteristic for students, and to some extent students are proud with those lifestyles and being different from other people with their different conerns:)
This word can be used sometimes in a way similar to ученик: namely, "ученик университета" is NEVER used, only "студент университета".
A professor can say both "она моя ученица" and "она моя студентка".
The former means something mores personal: I teach/I've taught her. I mean: i can say this thinking of how I'm explaining stuff to her. Or how we both are proud with that. A personal aspect of teaching, like with Christ.
THe latter means our "formal" status with respect to each other: I work in universtiy, she studies there, she's in my class.
Of course the latter implies some of the former, and the former may imply some of the latter. But accents are different:)
There's major difference between English and Russian usage! When English speakers introduce themselves formally in Russian Notebook here, they can write "я - студент". A Russian would not do so!!!
A Russian would write "Я учусь в университете" to define his/her occupation now. Young people often hear the same (a bit silly:)) question: "ты работаешь или учишься?"
A bit clumsy, becase 1. why this is important? 2. why these two options:) and it is asked way too often:)
"Я - студент" sounds like "I drink beer, play video-games, start preparing to exam sessions too late, and 2 month ago I kissed a girl/boy for the frist time! I'm mostly happy but I have hangover."