To attempt to explain it in the simplest terms, a continuation is a
representation, at a particular point in a program, of
everything the program is capable of doing subsequently. A
continuation is a potential that depends on initial conditions.
Rather than loop in a traditional way, it is possible to invoke
the same continuation recursively with different initial
conditions. One broad claim I have read is that
continuations, in a theoretical sense, are more fundamental and
underlie every other control structure. Don't worry if these
ideas cause your brain to melt; that is a normal reaction.