Scale degrees are the names given to each note of a scale. They can be given as numbers (usually written with a little upward-carrot above them), solfege (not covered in this node), or proper names.
The first note of the scale is scale degree one, and the ascending scale corresponds to each number (therefore in C, scale degree 1 is C, 2 is D, 3 is E, etc.). When Roman numerals are given to the scale degrees, they refer to the chord built off of that degree. So IV is the chord built off of scale degree 4. The quality of the chord (whether it is major, minor, augmented, or diminished) depends on the key signature (unless there are accidentals). Roman numerals show quality depending on their case: uppercase numerals mean the chord is major, and lowercase means minor. Adding a plus symbol to the right of an uppercase numeral means it's augmented, and adding a degree symbol to the right of a lowercase numeral means it's diminished.
Scale Degree Name Roman Numeral
1 tonic I in major, i in minor
2 supertonic ii in major, ii dim. in minor
3 mediant iii in major, III in minor
4 subdominant IV in major, iv in minor
5 dominant V in major, V or v in minor
6 submediant vi in major, VI in minor
7 leading tone vii dim. in major, VII in minor
8 is an octave away from 1 so there is no need to rename it.
The names given to each scale degree are often used to describe not just the note itself but the entire chord built off of it.
It should be noted that in natural minor, there is no leading tone. The name leading tone applies to the scale degree that is one half-step from tonic, but in natural minor, the 7th scale degree is a whole step away. In this case it is just referred to as the seven (not seventh, which refers to an additional third stacked on top of a triad, i.e. C E G is a triad, C E G Bb is a seventh chord where Bb is the seventh).
Note: this node is specific to major and minor modes only.