A server left intentially open or made easy-to-crack for the purposes of security research. From this vantage point, crackers' techniques can be observed and learned-from. Honeypots are often used to take the fall for a DDOS, watch efforts made in a select area (such as a university network), or simply to observe general types of attacks carried out against other servers.
The term comes from another biological equivilent, much like "virus" and "worm." It refers to an area which is sweet and desirable to an animal or insect that attracts visitation and allows the creature to be observed.
Another definition is available here.