Because this form of mathematics deals with entities (usually sets) which are, in essence, finite and non-continuous, discrete mathematics may be easily juxtaposed with other forms of mathematics such as calculus, which might be considered a form of continuous mathematics.
Continuity vs. Non-Continuity
The debate over whether or not the physical universe was continuous or non-continuous in nature goes all the way back to ancient Greece, when some of the first thinkers began to ponder these ideas, leading Democritus famously to suggest that matter itself might be non-continuous - that by "chopping" up matter into smaller and smaller pieces, one would eventually be able to reach the end, arriving at the smallest possible unit - the "atom."
Where the science of calculus assumes otherwise, foregoing the idea of distinct, definite values for points in space or time in favor of mathematical exact "estimates," discrete mathematics takes the opposite route, only dealing with sets which have "countable" elements.
The question still remains in terms of physics: Is the universe essentially discrete, or is it continuous? The laws of quantum mechanics throughout the 20th century have been leading many physicists to believe that both time and space are essentially discrete - that by cutting up these dimensions into smaller and smaller quantities, eventually they will be able to find the smallest possible unit (just as the photon is seen to be the smallest "discrete" amount of possible energy). This is the basis for several physical theories, most famous among these, perhaps, is string theory.
Discrete Applications
As mentioned previously, discrete mathematics tend to be focused today mostly in computer science, forming the basis for programming languages and circuitry encoding, which means that today, discrete mathematics is more important than ever before.
Where calculus arose in the late 17th century as a result of the need for a mathematics which explained the non-continuity which was found in nature, discrete mathematics has arisen out of the need for math which would provide practical applications for self-sustaining (and even learning) man-made electronics. Binary operations, Boolean logic, set theory, game theory... any number of today's most popular mathematical subjects are forms of discrete mathematics.
In fact, any sort of mathematics which relies heavily on algorithms must surely be based on discrete mathematics; a subject which continues to grow and expand as humanity finds ever greater uses for it as modern technology continues to advance at an excitingly rapid pace.
