Bobbin Lace is a delicate lace that uses wound spools of thread (the bobbins) to weave together the shapes in the lace. The lace is held in place with very fine lace pins while being worked and is made on a lace pillow (a cloth form traditionally stuffed with straw or sawdust).
Pairs of bobbins are twisted or braided to form meshes (also called "ground") or woven to form solid shapes, depending on the type of lace made.
Many styles of lace were made in the heyday of lacemaking (AFAIK that stretched between the 1500s-1700s) before machine lace became available.
The advent of machine lace at first pushed lace-makers into more complicated designs (ones that the machines couldn't handle) and then eventually pushed them out of business almost entirely. The resurgence of lace-making is a recent phenomenon and is mostly confined to a "hobby" status these days.
There are a few websites dedicated to the art of bobbin lace-making, one of the best being: http://www.bobbinlace.net which provides a good source of images and general information to show the differences between the various types of bobbin lace.
Some well-known types are: