Have you asked your teacher about scales? Has your teacher said anything about them?
Scales pretty much form the set of "correct" notes to play in a song. (Composers put other non-scale notes in there, but usually not many, and they draw attention to themselves because they sound different.) Different scales have different sounds, like C major and A minor, which both use only the white keys but start on different ones.
All major scales sound the same, except for the starting note. If you start on D, then you need to use an F-sharp and a C-sharp in order to preserve the intervals from one note to the next. That's easy to figure out on the piano keyboard because each adjacent key is a semitone away from the next one, whether black or white. From E to F is a semitone even though they are both white, so if you want the same interval as C to D, but starting on E, you need to go to F-sharp to get a whole tone. Same thing for any scale when you start on a different key. Knowing that, if you want to practice scales you can figure them out from what you know (or download a circle of fifths or something).
Scales are pretty much developed around the chords that a composer wants to play. Chord progressions determine the basic sound of a song. For instance, C-C-F-C. If you just arpeggiate the notes in those chords and end on a C in any octave, it will sound like a simple song. If your melody contains just the notes in the chords it may sound boring, but it will sound like music. If you use the A-A-D-A progression with the white keys it will sound different even though you're still playing the white keys. It's the chord progression that makes A-minor different from C-major.
Why practice scales and chords? When you see a key signature you won't have to figure out which keys it means. Many parts of songs are sections of scale or chord, and it will help your sight reading if you recognize the patterns.
A lot of piano players don't seem to put a lot of value in practicing scales. I'm guessing they don't do a lot of sight-transposing. Some musicians have to transpose routinely, it depends on what they're doing. When I was young I used to wonder why an arranger in a method book would, for instance, put a B-flat in the key signature when there were no B's to be played in the piece. Then I tried playing a song transposed up a tone and it didn't sound right. There were no B-flats, but the transposed score had F-sharps, which I would have known if I had considered that I was starting in F, and up a tone means G, which has an F-sharp. If you know your target scale well that saves your brain a lot of processing while you transpose. Scales and chords are also important for improvisation, which is big in jazz but exists in all forms of music.
What to practice, scale-wise? Not everyone sees it as being that important. If you're interested in it, download scales and chords, or buy a Hennon, and practice them all day. If you're just interested in playing a piece of music as written but have trouble remembering which black keys to hit, practice that scale until you're comfortable with it. Plus learn the blues scale and dink around with it because it's just fun-- pretty much any sequence of notes will sound like the blues, and simple improvisation is easy in it.