The very first thing you should learn about intervals is whats called enharmonics. Enharmonics are the exact same pitch, just spelt differently. For example, "C"-sharp is the same key on the piano as "D"-flat. Why are they spelled differently if they sound the same? The answer will eventually become obvious as you learn. Keeping in mind the differences with "E" and "B" from the previous paragraph, "E"-sharp is the enharmonic of "F". The note "C"-Flat is the enharmonic of "B".
The next step is to figure out what quality the interval is. There are five types of qualitys: Major, minor, augmented, diminished, and perfect. The chart below refers to figures 1-12 to show how many half steps there are in the major, minor,and perfect intervals.
If a perfect or major interval is made one half step larger (without changing its interval number) it becomes augmented. If a perfect or minor interval is made one half step smaller (without changing its interval number) it becomes diminished. An augmented fourth (and it's enharmonic - a diminished 5th) is called a Tritone.
Enharmonic Intervals are intervals with the same sound that are spelled differently.
If you were to play the above example on a piano, you would find that you would be playing the exact same notes each measure. They all sound the same, but are spelled differently. The reason for spelling them differently becomes obvious when you learn more about theory. Care must be taken in spelling intervals. If a specific interval is requested, the enharmonic-equivalent spelling is not correct. Thus, if a major third above E-flat is called for, a diminished fourth above D-Sharp is not correct, even though the sound is the same.
The inversion of an interval means that the lower tone of an interval becomes the higher tone, or the higher tone becomes the lower tone.
Intervals greater than an octave are called compound intervals. These intervals are named in a similar manner to the intervals within an octave (simple intervals).
The compound intervals are often labeled as their simple evuivalents-as if an octave were removed from the interval. The compound names are used only if it is important to stress the exact interval size.