Discrete mathematics is a branch of mathematics used extensively in computer science. It involves solving problems with finite data sets and a finite number of possible solutions. The goals of discrete mathematics have as much, if not more, to do with the development of processes, called algorithms, used to solve problems as they do with finding specific solutions. Discrete mathematics considers algorithms not just for their ability to find solutions, but for their efficiency and their ability to be executed by computers.
Discrete

The fundamental difference between problems in discrete mathematics and problems in calculus or algebra is the kind of data the problems deal with. Calculus and algebra deal with information on an infinite continuum, but discrete mathematics deals with "discrete" information, which is independent and finite  like bits of data are in computers. A simple problem might ask how many 4digit numbers can be made using only the numbers 1, 2 and 3. Such a problem would be solved by considering the possibilities for each digit individually.
Algorithms

Discrete mathematics deals fundamentally with the development of algorithms. Algorithms are explicit, stepbystep procedures for performing calculations or solving problems. An example procedure for finding the number of possible 4digit combinations of 1, 2 and 3 would be to first count the possible values for the first digit of a 4digit number composed of 1's, 2's and 3's, to repeat this for the next three digits, and to multiply the possible values for each digit  3 3 3 * 3  for the solution, 891.
Complexity

Discrete mathematics is concerned with more than just developing algorithms, however. One of the goals of discrete math is to make sure finding solutions doesn't take more work or time than is practical or permissible. For example, finding how many 4digit numbers can be made using only 1's, 2's and 3's can be done by listing and then counting all possible 4digit combinations of 1, 2 and 3, but such a solution would take far more work than considering the digits individually and multiplying.
Computers

Because discrete mathematics is most used in computer science, its ultimate goal is to allow computers to solve problems. This highlights the importance of algorithms. Because computers can only follow instructions  with no understanding of what they're doing or why  those instructions, which are algorithms, must be precise enough to find the solution and simple enough to not use excessive processing power or physical storage.
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References
 Discrete Mathematics, Fourth Edition; John Dossey, et al.
 University of Florida: Data Structures & Algorithms: Review of Discrete Math
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