Named after Galileo Galilaei - Italian astronomer, physicist (1564-1642).
Galilaei is a lunar impact crater located in the western Oceanus Procellarum. Some distance to the southeast is the crater Reiner, while to the south-southwest is Cavalerius. Northeast of the crater is a meandering rille named the Rima Galilaei. To the southeast is the unusual Reiner formation, a swirling arrangement of light-hued ray-like material. (wikipedia entry)
A crater is a circular depression likely created by an impact event. On the Moon they are named after deceased scientists, polar explorers, astronauts or cosmonauts.