Bending a note is a wonderful effect - much used by jazz musicians - but by no means restricted to them.
Physically speaking, bending a note means to approach that note through a chromatic succession, going through all the microtones. For instance, to bend a D, you would start on the C immediately below and go through a "continuum" of sound until you reached the D.
Unfortunately, as Prometheus said, this cannot be done on a normal piano, because there is no continuum of microtones to go from C to D. All you have is a semitone (D#) interval between C and D.
This is not true of wind and string instruments, where you do have the possibility of creating a continuum of frequencey form one note to the next, so bending notes is really at home with these intruments.
However, it is possible to create the "illusion" of bending a note, by - as Bob wisely mentioned - crushing two (or more) consecutive keys on the piano. for instance, by starting on C and playing very quickly the C-D#-D. Most of these short embellishments in jazz are attempts at reproducing the bending effect on the piano.
But from what your teacher said, and from what you youself said (I´m hitting the keys in time with the metronome") I believe what he means is indeed "swing" (well done again. Bob!

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There is a convention in jazz notation that 2 quavers are to be played as a triplet with the first two notes of the triplet tied. So that the first quaver is slightly longer than the second: the first quaver would be together with metronome beat, but the second would be slightly after the second beat. Even the triplet notation is an approximation of the real thing. One has to learn to swing by listening a lot to performers who are good at it. Mathematically figuring out a notated score will not help.
Best wishes,
Bernhard.