In music, fingering means, basically which finger you should play a note with. Fingering according to the music is technically optional, most fingering is written by the editor, not the composer. However, it can sometimes be very difficult to play some passages without using the proper fingering. In playing the piano, each finger has a designated number, shown here:

In most pieces of music, there will be small numbers above notes in tricky passages indicating which finger should be used to play the note. Consider this short excerpt from Mozart’s Third Piano Sonata:

The small numbers above the notes (below for the left hand) indicate which finger should be used to play each note. While it’s okay to use different fingers, it would be nearly impossible to play this passage any other way. So as you’re learning a new piece, pay close attention to the indicated fingering and don’t practice any other way. If you learn the wrong fingering, it can be very difficult to correct later. Nothing will screw up a run like running out of fingers before you get to the end.