A "mole" in science is a unit that is used for measurement, and is often referred to as "Avogadro's number." This is because Amedeo Avogadro, an Italian chemist and physics professor, proclaimed in the year 1811 that gases with the same volumes at the same temperature had an equivalent quantity of molecules in its composition. This theory was developed even further fifty years later, with an Italian scientist named Stanislao Cannizzaro. He used Avogadro's original hypothesis to create a set of atomic weights for the known elements by performing comparisons on the masses of gases with equal volumes. In the year 1865, Austrian school teacher, Johann Josef Loschmidt further the research efforts of his predecessors, and was successfully able to calculate the size of an air molecule. This was significant because it allowed for an estimation for the quantity of molecules in a given volume of air. All of these efforts led to the mole, which can be defined as a theory that in a certain mass of an element, there is an exact number of atoms.
The mole is important for scientists because it allows for the quantities of microscopic things, such as atoms and molecules, to be calculated in an easier fashion. Like the term "dozen" is used to simplify a quantity of twelve, the mole allows for extremely small decimal-pointed numbers to be expressed in an simpler way. If you have a periodic table, you can convert moles into grams for different elements. The mole also represents the number of particles of a certain substance.
Sources:
http://www.enotes.com/science/q-and-a/what-mole-chemistry-287632
http://www.visionlearning.com/library/module_viewer.php?mid=53
