Diophantine equations are equations for which solutions are restricted to a certain class of numbers, such as positive integers, negative integers, rational numbers, or some other criterion. They are named after Diophantus, a Greek-educated Egyptian who lived in Alexandria sometime between 150 A.D. and 350 A.D. and who investigated several types of Diophantine equations.
In general, Diophantine equations can be quite, um, interesting. There are not a lot of methods that can be applied to solving a wide range of Diophantine equations; each type of equation has its own methods of solution. As an example, Diophantus showed how to find all solutions in natural numbers for x² + y² = z² (see Pythagorean triples), but there are no solutions in natural numbers for x³ + y³ = z³, or for xn + yn = zn, where n is an integer ≥ 3 (see Fermat's Last Theorem). Also, although the equations x² + y² = z² and x² − Ny² = 1 (see Pell's equation) look quite similar, the methods used to solve them are very different.
In 1900, the German mathematician David Hilbert presented a list of 23 unsolved problems that he believed to be significant. Hilbert's tenth problem asked for an algorithm that would allow us to tell whether a given Diophantine equation had solutions or not. In 1970, Matusevic proved that such an algorithm is impossible.
There is a worksheet on Linear Diophantine Equations available on Math Lair.
Sources used (see bibliography page for titles corresponding to numbers): 20.