A circular definition is one that assumes a prior understanding of the term being defined. For instance, we can define "oak" as a tree which has catkins and grows from an acorn, and then define "acorn" as the nut produced by an oak tree. To someone not knowing either which trees are oaks or which nuts are acorns, the definition is fairly useless.
A circular definition occurred in an early definition of the kilogram. The kilogram was originally defined as the mass of one litre of water at standard pressure and the temperature at which it is densest (which is about 4°C). The unit of pressure is the newton per square metre, where a newton is the force that accelerates a kilogram one metre per second squared. Thus the kilogram was defined in terms of itself. It was later defined as the mass of a certain piece of metal in Sèvres.
A circular defintion also crept into the classic defintion of death that was once "the permanent cessation of the flow of vital bodily fluids," which begged the question on what made a fluid vital.
A branch of mathematics called non well-founded set theory allows for the construction of circular sets. Circular sets are good for modeling cycles and, despite the field's name, this area of mathematics is well founded. Computer science allows for procedures to be circularly defined by using recursion. So even two very precise fields of study allow for circular definitions of objects to exist.