I diode is the electronic version of a one way valve in plumbing. It allows current to flow in one direction but not the other. How it does this is basically magic, but if you must know, study semiconductors and semiconductor junctions first.
Like all things, diodes are not perfect: There will be a slight flow of current backwards before the diode kicks in and stops it. They are also limited: If you push hard enough (high enough voltage) the diode will give up and conduct in the backwards direction. They also will not conduct in the forward direction until enough voltage has built up to activate the diode; the forward voltage drop. All of these imperfections are different for different types and grades of diodes. Read the datasheet.
The Light Emitting Diodes, or LED is a type of diode.
Which is anode and cathode? Easy way to remember this, look at the diode
symbol and you will see that part of it forms a K, which stands for Kathode
(nay, Cathode). Just remember not to spell it like that when you write it
:) i.e.
(K)athode Anode ---------|<------
Thanks to Michael Rigby-Jones for that!