Listserv is software for managing mailing lists: creating lists, distributing emails over the list, adding new subscribers, adjusting settings. It can be operated via web interface, by sending email commands or via TCP.
liblistserv is the glue between listserv and your C-programme for manipulating or querying mailing list. By calling the provided functions, your programme does not need to parse the results. Liblistserv is a reentrant library, written in C, meant for communication over TCP with a running Listserv instance.
Check the manual for more details.