Euclid defines a circle in Book I as a plane figure contained by one line, such that all straight lines drawn from a certain point within it to the boundary are equal to one another. That point is the centre. The definition is a condition on distance, and everything else about the circle follows from it.
His third postulate then grants that a circle may be described with any centre and any distance. This is the compass, stated as a permission rather than as a tool, and it is one of only five things Euclid asks to be granted. Together with the straightedge of the first two postulates it is the entire apparatus of classical construction.
There is a subtlety about which compass. Euclid's postulate has often been read as describing a collapsing compass, one that closes the moment it is lifted and so cannot carry a distance from one part of a diagram to another. Euclid does not need the stronger tool, because Propositions 2 and 3 of Book I show how to transfer a given length to a given point using only the collapsing kind. The result is that the two compasses are equivalent in power: anything the rigid compass can construct, the collapsing one can construct too, at the cost of extra steps. It is an early and elegant demonstration that a restriction can cost nothing at all.
The circle is also the source of nearly every new point in construction work. Bisecting a segment, erecting a perpendicular, copying an angle, bisecting an angle: each of these standard moves is done by striking arcs and using where they cross. The straightedge joins; the compass decides. When a construction turns out to be impossible, it is because no finite sequence of such crossings can reach the required point.