What is the difference between `using` and `import` in Julia when building a module?

The Julia Modules documentation states:

The import keyword […] only
operates on a single name at a time. It does not add modules to be
searched the way using does. import also differs from using in that
functions must be imported using import to be extended with new
methods. […] Functions whose names are only visible via using cannot be extended.

(Emphasis mine.)

For example, you can use import to add methods to Base.show to pretty-print your own types, but not with using.

There is also importall that imports all exported names of a module.

(This answer refers to Julia 0.6; the documentation was reworded for 1.0.)