Intervals are deceptively easy little things. To define them, they sound extremely rudimentary: intervals are simply the distance between two notes or pitches. Play a note on your piano. Now play another one. The distance between those two notes is the interval. Upon closer inspection, however, we see that intervals are far more complex than that. There exist several different types of intervals, all of which can be altered in a number of ways, and understanding the nature of these changes is at the very core of a solid music theory education.
Intervals are defined or named by two characteristics: the interval's number and the interval's quality. The interval's number describes the number of staff positions that sit within the intervals: a first, a second, a third, etc. For example, the interval number for a C and an F would be a fourth because there are four notes between those two (including the C and the F themselves). Likewise, a C and an E would be a third, a C and a D would be a second, and so on.
The interval's quality is a bit more complicated. Interval quality describes the specific type of the intervals in addition to the number; intervals can be perfect, major, minor, augmented, or diminished. But not every interval can be of every quality. While all intervals can be augmented or diminished (by adding or subtracting a half step, respectively), only unisons, fourths, fifths and octaves can be perfect; a perfect fourth is five half steps, a perfect fifth is seven, and a perfect unison is zero (since a unison represents the same two notes). Similarly, only second, third, sixth, and seventh intervals can be major or minor; like augmenting or diminishing, this is achieved by adding or subtracting a half step from the intervals.
But be careful. Since major and minor intervals are created by altering the intervals by a half step, augmenting and diminishing works a little differently here. Instead of simply adding or subtracting a half step, augmented intervals (in this case) are a half step more than the interval's major, and diminished intervals are a half step less than the interval's minor. Let's consider third intervals, for example. Major third intervals are four half steps and minor third intervals are three; in a fourth interval, that half step down that creates the minor would create the diminished. But for these intervals we have to go a half step below that, making a diminished third two half steps (which actually creates major second intervals, but that's a story for another time).
An interval is the distance between any two notes, and in this course you are going to learn to hear 2nds, 3rds, 4ths, 5ths, 6ths, 7ths, 9ths, 11ths, 13ths, and of course octaves & unisons. You will also learn to recognize them in printed music quickly. 
