Lichens are actually two different organisms living together. One is a fungus, and the other can be either an algae or a cyanobacterium. Since it is made up of two different organisms, how do lichens reproduce?
New lichens are formed in two different ways. When they are dry, they become brittle, and it is easy for pieces to break free. Those pieces can be carried by wind, water, animals, etc. If the pieces come to rest in a good spot, it will grow into another lichen.
The fungus part of the lichen also reproduces by producing spores. If a spore lands in a good spot, and if there are either algae or cyanobacteria of the right type in the same spot, they will join together to form a new lichen.