Our friend the interval.
First, look at the name of the two notes. (The following assumes that all the intervals are within an octave.)
Note names are the letters, and the note names wrap around. Starting with an A, the note names are A, B, C, D, E, F, G, A, B, C, D, E, F, G, A. Etc.
If the first and second note have the same name and they're not an octave apart, it's a unison.
If the higher note is one note name higher than the lower note, it's a second.
If the higher note is two note names higher than the lower note, it's a third.
If the higher note is three note names higher than the lower note, it's a fourth.
If the higher note is four note names higher than the lower note, it's a fifth.
If the higher note is five note names higher than the lower note, it's a sixth.
If the higher note name is six note names higher than the lower note, it's a seventh.
If the higher note name is seven note names higher than the lower note, it's an octave. (And the note names are the same.)
OK, now you know what the size of the interval is. But not all seconds (thirds, fourths, etc.) are the same.
If the upper note is in the major scale of the lower note (and the major scale IS defined only one way), then the interval is a major whatever. If the lower note is in the major scale of the upper note, then the interval is a minor whatever. If the upper note is in the major scale of the lower note AND the lower note is in the major scale of the upper note, it's a perfect interval. (That's why the fourth and the fifth are often called perfect intervals.)
BTW, the major scale is defined a whole step, whole step, half step, whole step, whole step, whole step, half step. On a keyboard, a half step is from one key to the next key (C to C#, etc), and a whole step is from one key up two keys. You include the black keys in your count. Not all white keys have a black key between them.
OK, now you've defined the perfect, major, and minor intervals. However, this doesn't include all possible intervals. First, remember that a minor interval will be smaller than a major interval. (Again, this is why perfect intervals exist, because the minor and major interval is the same.) If you go down a half step from the minor (or perfect) interval, you've got a diminished interval. If you go up a half step from the major (or perfect) interval, you've got an augmented interval.
What makes it interesting is that identical sounding notes can have different note names. In simple terms, a Bb is the same key (note) as an A#. A Bb to an F is a perfect fifth. (An F is in the key of Bb major, and a Bb is in the key of F major.) But an A# to an F, even though it sounds identical on a modern keyboard, is a diminished sixth.
All these interval names make a lot more sense in the world of classical music theory, where the note names exist for a reason based on the key of the piece, not just which key is being struck.
Going back to your question (and we've gone a bit afield here), a minor third is a third in which the lower note is in the major key of the upper note. An augmented second sounds exactly the same, but is spelled differently. Counting half steps, a minor third is three half steps from the bottom to the top note. (And so is an augmented second.)
Here's a quick (badly formatted) table, with the first column being the bottom note, the second column being an augmented second, and the third column being a minor third. (You'll notice that the second and third columns sound the same).
A B# C
A# Bx C#
Bb C# Db (yes, this is the same as the previous line)
B Cx D
C D# Eb
C# Dx E
Db E Fb (yes, this is the same as the previous line)
D E# F
D# Ex F#
Eb F# Gb (yes, this is the same as the previous line)
E Fx G
F G# Ab
F# Gx A
Gb A Bbb (yes, this is the same as the previous line)
G A# Bb
G# Ax B
Ab B Cb (yes, this is the same as the previous line)
(And this brings us back to Dough.)
x is the symbol (sort of) for a double sharp. You can get worse with this, but this is probably enough. I also skipped a few, like B#, Cb, etc.
(Watch for typos.)
Does this help with the minor third/augmented second?
