Principles of operation

An electron tube consists mainly of a source for electrons, some form of control elements and an anode. All this is placed in a container of glass or metal that is filled with vacuum. The electron source can be a directly heated filament  or an indirectly heated cathode. The electron flow emitting from the filament or cathode is, except for the diode,  controlled in some way by means of electrodes with varying potentials in order to attenuate or accelerate the flow of electrons. The electrons are collected at the anode which usually carries the highest potential with respect to the cathode.

Although the electrons flow from the cathode to the anode, it is generally accepted that current flows from the anode to the cathode.

The cathode are marked c or k in schematics and the anode is marked a. The grids are marked g or when more than one grid is present they are numbered g1, g2 etc, with g1 closest to the cathode.

Below are some descriptions as well as examples of different electron tubes.
Basic forms of electron tube systems
Name and schematic symbol Description Examples

 Diode

The simplest form of an electron tube. Since there is no control elements current flows without limitation in one direction. Used for signal detection and rectification of alternating voltage in power supplies. The symbol at right is a duo-diode that can be used as a full-wave rectifier. Signal detectors: EAA91
Half wave rectifier: CY1, DY86
Full wave rectifiers: AZ1, GZ34
Triode
A control grid is placed between the cathode and the anode. Thus it is possible to control the flow of electrons. The triode is the most common tube type and is used both for small-signal amplification as well as high power transmitters. A gas-filled triode is often called Thyratron. HF low power. EC92
Audio output: 2A3, 300B
Transmitting tube:
Tetrode
A second grid called screen grid is placed between the control grid and the anode.
In the early days the extra grid was in some cases placed between the cathode and the control grid and was ..
Transmitting tube:
Pentode
The third grid is placed between the screen grid and the anode. This grid is called the suppressor grid. Small signal amplifier: EF86
Audio ouput: EL84, EL503
Transmitter output:
Hexode
- -
Heptode
- -
Octode
- AK1
Nonode
- Only known tubes are the EQ40, EQ80 and UQ80 used for FM-detectors

The functions above have been mixed in a great variety of combinations.
Here are some examples:

Diode+Duo-diode+Triode
- Detector and LF-amplifier PABC80
Triode-Hexode 
Used as oscillator and mixer in AM-receivers. Triode-heptodes are also used for this purpose. ECH81
Some special tube functions:
Secondary emission tube EFP60
Beam-deflection tube. Used as a synchronous demodulator in color-TV sets. 6AR8, E80T
Magic eye tube. Has a fluorescent screen indicating the level of a received signal or the audio level in tape recorders. EM34, EM84EMM801 etc
Simple cathode ray tube of triode type for magnetic deflection. Used mainly in TV-sets and for radar displays. MW6-2
Simple cathode ray tube for electrostatic deflection. Used mainly in oscilloscopes. D8-11BG
If you omit the heater and fill the glass bulb with some form of gas you will get these types of tubes:
Gas discharge tube. When the voltage between the electrodes reaches a certain value the area around the cathode will lit up with an orange glow. Voltage stabilizer: 150C1
Voltage reference: 85A2
As above but with several cathodes. Here the cathodes are formed as digits and placed so that they are visible from the top or the side of the tube. These tubes are commonly called Nixies. Digit indicator: ZM1000, ZM1020
Various signs: ZM1023
Multi segment: B7971
And a lot more HERE.
Here a voltage is applied via a resistor across the electrodes. Depending upon the design and the gas used this tube type can be a photoconductive cell where the voltage between the electrodes varies in proportion to the light applied onto the specially treated photo cathode. Or a radiation counter tube that is used for measuring alfa, beta and/or gamma radiation. Photocell 3547
Radiation counter 18514

And there are many other strange tubes developed through the years.
Look at my list of Fancy Names to see some more.