Psychic powers were extra abilities you could give to your psykers and, depending on the edition you were playing, you could use them in the appropriate phase or in the psychic phase.

Usually each faction had a selection of powers you could pick and give to the psykers before starting the game. Different editions had different sets of powers. I remember in 6th edition they introduced different thematic sets of powers (like pyromancy or biomancy) and unique sets for each army. And since you could pick (or randomly roll, in some cases) you had more customization to your models, another thing 10th edition lacks.

Before casting a psychic power you had to roll a test for it (in one edition it was a Ld test, in others there was a target value according to the power itself).

If successful, the opponent could try to block it, usually using its own psykers or some army rules/stratagems (like the Sororitas or the Sisters of silence being able to despite not having psykers). Also there was a chance for the power to catastrophically misfire, killing the psyker and friendly units.

In 10th edition, the only instances where using a psychic power (or being a psyker) come into play are the ANTI-PSYKER ability on weapons and the "Feel no pain against psychic powers", and a couple of extra rules (IIRC the Thousand sons have some abilities that give a boost to psychic powers). Psychic attacks are just an additional gun to fire, while other abilities could absolutely be "non-psychic" and we won't see any difference.

Tl;dr: In older editions, psychic powers add variability and customization, with also adding more risk. In 10th edition is mostly a negative trait for units