I'm a maths graduate, I'll give this a try.
Basically it's an unfortunate side effect of how maths has evolved from year dot. People didn't sit down and work out a set of axioms wrought through heavy analysis. They simply went with things like 1+1 = 2, etc - analysis came thousands of years later.
Hence we have situations like the square root. The square root of a number usually has two roots and both are real numbers. Often in science and physics we need some way of distinguishing between them, for example wave functions.
This is where the term 'imaginary' comes in. They are not literally imaginary numbers. It's just a way of getting past the "poor" way mathematics was designed. Note how I used the word usually above - believe me when dealing with logic it gets frustrating seeing that word crop up.
Now I'm sure a maths professor would get angry and what I've said here, but it does hold some merit.
If you want to look further into it, pick up a book on Mathematical Analysis.