Complex numbers consist of two components: a real component and an imaginary component.
The imaginary component always involves the imaginary square root of -1 and it's usually given the symbol i or j (in electrical engineering to distinguish it from i for electric current).
Complex numbers add a new dimension to numbers: while "ordinary" (real) numbers can be represented on a number line, a complex numbers need a plane to be represented, so that one dimension of the plane represents the real part and the other dimension represents the imaginary part.
For some problems complex numbers need representation in three dimensions.
This extra dimension enables the use of complex numbers to address a variety of problems, especially in engineering, in a way which is relatively easy to manipulate. These problems typically include treating variables which can be expressed as the components of complex numbers. Complex numbers can be used in branches of mathematics to solve problems (for example, in trigonometry) which would otherwise be more difficult to solve or take longer to solve.