Most, but not all scales have scale degree numbers of 1 through to 7. The starting note is known as 1 and the numbers are assigned in order, so the 7th degree or note of a scale is the one just before you get to a higher version of your starting note.
So the scale degree names are (which are usually written in roman numerals):
I is the Tonic
II is the Supertonic
III is the Mediant
IV is the Sub-Dominant
V is the Dominant
VI is the Sub-Mediant and
VII is the Leading Note
You should notice that the V or 5th note is called the dominant and the IV or 4th note is called the sub-dominant. This is no accident or coincidence as the V note is 5 notes about the tonic and the IV note is actually 5 notes below the tonic. Both are considered to be almost as important as the tonic note.
Likewise, the III or 3rd note is the mediant, while the VI or 6th note is known as the sub-mediant. This is because the the III note is 3 notes above the tonic, while the VI note is 3 notes below the tonic. Both the mediant and the sub-mediant notes are considered equally important in both major and minor scales.
The VII note is called the Leading Note because it leads towards the tonic note. When the leading note is played within the context of a piece of music, it can often be heard leading or leaning towards the tonic – it often literally sounds like it is pulling towards the tonic.
All of notes of the scale are important and do provide a level of interest and variety to pieces of music, but there is a hierarchy of importance and it is:
Tonic
Dominant and Sub-Dominant
Mediant and Sub-Mediant
Supertonic and Leading Note
Even though this comes from a traditional theory framework, it is actually quite useful if you are playing the piano with a chord approach. If you are playing the piano using chords, then the more you know about chords, the more you can get creative with the chords you play.
So using the above information you could work out which chords you could say play two of the base note of the chord, or maybe the middle note, depending on the key of the piece of music and the context. Take for example the key of C major. If you were supposed to play an F chord, which in this context is the sub-dominant, you are playing an F, A & C. Knowing that your F is the sub-dominant and C is the tonic, you could freely double up on the F note or the C note or even both. If we then looked at this same chord consisting of the notes F, A & C, this chord can also be found in A minor and in this context it would be based on the sub-mediant chord. The note A is not only the middle note of the chord, but in this context, it is actually the tonic note in this particular piece and therefore a very useful note to play more than one of when playing the chord.
While it can get a little tedious trying to keep all the numbers and letters that are involved in working out chords and the notes that they contain, having the scale degree notes having actual names, helps to keep things clearly defined. I would thoroughly recommend learning them and committing them to memory.