Aside from people's very worthwhile answers describing the complex number system, I think it is worth mentioning that the use of the term "imaginary" is an unfortunate historical remnant. In experience, a lot of the average student confusion comes from their trying to get their head around the naive meaning of imaginary.
Now that modern mathematics understands that all number systems are more or less games with axioms, we know that no part of a number system is really more imaginary than any other part. "Imaginary" might better be termed "augmented" - we can augment the "real" number system by adding an element "i" which we say is equal to the square root of -1.
And it is just as ironic that the "real" number field itself has perhaps as many weird elements as say the rational "imaginary numbers" (pi, e, Theta etc)
Also ironic is that real numbers may not be "real" (i.e., exist) at all. The uncountable part which is THE part that completes rationals to continuity cannot be described or generated in any way since there are only COUNTABLY many different computer programs (or mathematical formulas).
Now that modern mathematics understands that all number systems are more or less games with axioms, we know that no part of a number system is really more imaginary than any other part. "Imaginary" might better be termed "augmented" - we can augment the "real" number system by adding an element "i" which we say is equal to the square root of -1.
And it is just as ironic that the "real" number field itself has perhaps as many weird elements as say the rational "imaginary numbers" (pi, e, Theta etc)