A remote function is a function in a module that isn’t this one.
A local function is one defined in the current module.

There is a slight cost in the BEAM to execute a remote function over a local one, but not material to a design decision.

A remote call is every function call which is qualified by its module.

defmodule M do
  def f(0), do: 0
  def f(n) when n > 0, do: M.f(n-1) + 2
end

The recursive call here is a remote call, with all of its costs.

That is not fully true as __MODULE__.foo() is also remote call. In short the difference between remote and local call is that you put the module name before remote call.