In translation, japa means repetitive prayer and mala means garland. The beads are used to count repetitions of prayers, chants or mantras to aid one's focus on the meaning or sound of the mantra. One repetition is usually said for each bead while turning the bead clockwise with the thumb.
108 bead malas are generally used by the Buddhist, Hindu and Sikh traditions. The 109th bead is the sumeru or guru bead, counting should start and end on either side of this bead. There are many styles of holding and using the mala and many types of mantras, therefore individual research is recommended to find the style that works for each individual.
A good mantra to get started with is Om Mani Padme Hum which essentially stands for universal compassion.