The official name is the Go Gopher.
However, it's very common for people within the Go community to simply call it the Gopher, as the context is usually understood.
Think of "Go Gopher" as its full name and "Gopher" as its universally accepted nickname. Both are correct in practice, but the former is more precise.
The Go Gopher was designed by illustrator Renée French, who also created Glenda, the Plan 9 bunny mascot.1 It was originally created for a T-shirt for the first Go conference.
One of the key reasons for its popularity is that the design is open-source, licensed under the Creative Commons Attribution 3.0 license.2 This has allowed the Go community to create countless variations and adaptations of the gopher for conferences, projects, and user groups.
It's important to note that the Gopher is the project's mascot, not its official logo.3 The official Go logo is the stylized, modern "GO" text.4 The Gopher, however, is the much more beloved and recognizable symbol of the Go community.