In this case, serial refers to the serial port, which is a (largely outdated) connector that allows for serial communication (which, in fact, implies not much more than a protocol that sends one bit at a time, through one dedicated wire). That stands in contrast to the parallel port which has multiple wires for sending and receiving. Don't let the amount of pins confuse you. Only two of the serial wires are used for data transfer, each in one direction (one for send and one for receive), as laid out in this pin diagram.
The Java Communications API provides ways for communicating with the underlying system's serial but also parallel ports, as you can quickly see from this documentation page. Since most modern computers don't have these ports anymore, this API might only be useful when running on certain embedded devices that still use these ports for external connections.
This HowStuffWorks article explains most asepcts of the Serial Port in layman's terms.
Let me further disambiguate "serial port" and "serial protocol". The serial port uses a serial protocol, but so do USB and Firewire. They also only have one transfer wire per direction (send and receive), but their protocol and technology allows them to transfer data much faster. Parallel protocols, in contrast, need more power and, as of 2013, there are few (I can't think of any, contributions welcome) that can match the speed of serial protocols, due to synchronization overhead.