Constant scale degrees are the 1st, the 3rd, and the 5th (in C major - C, E, and G, in A minor - A, C, and E). They are part of the tonic chord and they are quite stable as opposed to inconstant the 2nd, the 4th, the 6th, and the 7th scale degrees which are unstable. They lead to the closest constant scale degree - the 2nd to the 1st or the 3rd, the 4th to the 3rd or the 5th, the 6th to the 5th, and the 7th upward to the 1st.
Here are the names of the scale degrees:
1st - tonic
2nd - supertonic
3rd - mediant
4th - subdominant
5th - dominant
6th - submediant
7th - subtonic.
The distance between the 1st and the 3rd scale degree is a major 3rd in major keys, and a minor 3rd in minor keys.
When you look at your organ piece and want to figure out the meaning of the notes, it's good to think about each note in terms of scale degrees. But keep in mind that the piece might have quite a few modulations (excursions into other keys) so you have to think about scale degrees in a new key as well (for example C in C major is the 1st scale degree but in G major it's the 4th scale degree.
[HT to Andreas]
Practice ascending chromatic sequence in A major. The chords: I-V43-I6 (see video example below):