The Liver birds are two large and famous bronze metal birds on top of a building - the Liver Building - in the city of Liverpool, England. The building has the offices of an insurance organization. The birds are symbol of the city, although no birds really look like this.
The sound of the end of the word "liver" in the name of the bird is like "driver" not "river". The start of the word Liverpool ("liver") sounds like the word "river". This is a funny play on words.
There is also a British television sitcom with the same name that is about two young women who share a flat in Liverpool. The play on words is increased by a second British slang meaning of the word "bird" to mean young woman.
Many people in Liverpool (known as Liverpudlians) are proud of their city and of the Liver birds. There is a story that the birds will come alive if a truly pure woman walks between them. This will never happen! This is an example of the humor of the local people.