What exactly is a connection pool at a low level?

The idea of a connection pool is simple enough to understand: caching connections (say, to a database) to avoid the overhead of opening and closing connections. But what does opening (and closing) connections and caching them *look* like? I think of connection pooling as conceptually analogous to thread pooling. Thread initialization includes stack allocation, construction of the TCB, and then a context switch. (I may perhaps be glossing over some details here.) I'm clueless as to what connection initialization entails. Is there a connection control block? How can different clients reuse connections from the same shared pool? Naively, I would think that connections are specified by their end points.