Diodes vs. Triodes

diode means two and it means that it has two electrodes Kathode and Anode. electrones are being "shot" from Kathode and being attracted by Anode when the voltage is applied to these electrodes.
triode means three electrodes are in the vacume tube Kathode Anode and Grid. triode has to be fed with kathode and gride offset voltages and the small variation of the input Grid voltage will inflict the large variation of the anode. this event is called amplification.
tetrode means four...
penthode means five...
dual-triode means two triodes in one bulb...

To simplify: a diode, aka rectifier in tube terms, has two pins - one lets in the positive and negative ac voltage and the other lets out only the positive (or negative). This sort of creates a DC current and is the first step in the power supply process of creating a DC voltage to be amplified.
The triode, or amplifer, has three pins. One lets in the audio ac signal, the second "takes in" in DC current (from the diode) and the third lets out the amplified signal. This works in the same fashion as a xerox machine copying and enlarging a document: the original document is the audio signal at the first pin, the blank copy paper is the second pin and the third pin is the copy paper in the output tray with the enlarged copy.
This is really overly simplistic as tubes have more pins than the prefix "di" "tri" etc imply. The other pins are used to let in current to heat the tubes in order for them to work, and as Marakanetz stated above, some tubes have more than one set of electrodes in the same single tube.

Also, some tubes can be used for stuff they were not designed for - a triode with the grid unconnected (or connected to another element?) can be used as a diode, though I can't wonder why someone would do that. Also, things like Pentodes can be run as triodes (resistor between a couple of the grids).

In Simple terms: Diode = rectifier, used in power supply.
Triode = amplifier, used in gain stages.

-Ed

I. General-Introductory articles. All of these will help answer your question. #4 might be it if you only want to read one short article.

1) ✔ A Taste of Tubes:
http://www.anthemav.com/OldSitev1/frames/tubesfr.html
2) ✔Articles Section, The National Valve Museum.
http://www.r-type.org/static/.contents.htm
3) The Cool Sound of Tubes:
http://www.spectrum.ieee.org/select/0898/tube.html
4) How A Vacuum Tube Works:
http://www.svetlana.com/docs/tubeworks.html
5) Vacuum Tube Valley FAXs:
http://www.vacuumtube.com/FAQ1.htm
6) Vacuum Tube FAX:
http://home.earthlink.net/~busenitz/vac.html
7) A History of Tube Companies:
http://vintagetubeservices.com/page8.html

If you want one step more read this

✔ Ray Dall, Electronics Theory.com
http://www.electronicstheory.com/html/e101-37.htm
Some basics on circuits about as briefly as is possible. About 50 pages. An overview of how a tube works begins on page 37.

This comes from an outline of online tube articles I have put together. It is geared at the DIY crowd but has other stuff to. If anyone wants a copy drop me an email.

I remain,

Did anyone mentioned that Diode can be semiconductor of a type p or n? Not only tube!