The term statically linked means that the program and the particular library that it's linked against are combined together by the linker at linktime. This means that the binding between the program and the particular library is fixed and known at linktime—well in advance of the program ever running. It also means that we can't change this binding, unless we relink the program with a new version of the library.
You might consider linking a program statically in cases where you weren't sure whether the correct version of a library will be available at runtime, or if you were testing a new version of a library that you don't yet want to install as shared.
Programs that are linked statically are linked against archives of objects (libraries) that typically have the extension of .a. An example of such a collection of objects is the standard C library, libc.a.