There are a few different schemes used in the naming of vacuum tubes. These are the ones you're most likely to find when building or repairing audio equipment and guitar amps. The information contained in a tube number includes the proper filament voltage, and an arbitrary 'sequence number' which serves as a unique identifier for individual series of tubes.

On American-made equipment, you're most likely to run into the RETMA system. RETMA designations consist of the filament voltage, a sequence code, and number of elements in the tube. Also, some American tubes have "JAN" in the number, which stands for 'Joint Army Navy' - these tubes were originally made to meet military specifications, and were built under tighter quality control. They may also be more rugged.

For example, the 12AX7 dual triode, used a lot in the preamp stages of guitar amps, has its numbering broken down like this:

12 (filament voltage), AX (sequence), 7 (7 elements).

Filament voltage:
0 is a cold-cathode tube, probably, a voltage regulator.
1 - 0.1-2.0 volts
2 - 2.1-2.9 volts
3 - 3.0-3.9 volts
For others, the voltage number specifies a filament voltage of N.0 - N.9 volts. Thus, a 12 would indicate 12.0 - 12.9.

Sequence code: This will be one or two letters. When all the one-letter sequences were used up, two-letter codes were introduced. A one-letter code, or two-letter code that ends in, the letters U through Z, usually (but not always) is used for rectifier tubes.

Element count: This includes the filament. For example, the 12AX7 has one filament, two indirectly heated cathodes, two grids, and two anodes.

Often, there will be additional letters after this standard designation. These designations are as follows:
A: Controlled heater warmup time. Could also be used to indicate increased ratings/performance.
B: Increased ratings/performance.
C: Same.
G: Glass bulb
GT: Glass Tubular
W: Ruggedized version
X: Low loss ceramic base
Y: Low loss phenolic base

Also common on tubes in audio equipment is the European numbering scheme. The first character of the type number will also be a letter, to denote the filament voltage or current.

A: 4 volt
B: 0.18 amp (series)
C: 0.2 amp (series)
D: Less than 1.4 volt (series/parallel wiring)
E: 6.3 volt
F: 12.6 volt
G: 5 volt (parallel)
H: 0.15 amp (series)
K: 2 volt
L: 0.45 amp (series)
P: 0.3 amp (series)
U: 0.1 amp (series)
V: 0.05 amp (series)
X: 0.6 amp (series)
Y: 0.45 amp (series)

Second and subsequent letters - Type
A: Diode (not rectifier)
B: Dual diode
C: Triode (signal)
D: Power triode
E: Tetrode (signal)
F: Pentode (signal)
L: Pentode or tetrode (power output)
H: Hexode or heptode (hexode type)
K: Optode or heptode (octode type)
M: Tuning indicator
Q: Nonode
Y: Rectifier, half wave
Z: Rectifier, full-wave

After this, there are two or more digits to designate the base type and sequence.

First digit: Base
1: Miscellaneous
2: Miniature, 10 pin
3: International Octal
4: Miniature, 8 pin, B8A
5: Magnoval, B9D
8: Noval, B9A
9: Miniature, B7G

Sequence number: If this is even on a signal pentode or tetrode, the tube is a sharp cutoff type. Odd numbers designate a remote-cutoff type tube.

Duncan's Amp Pages: Vacuum Tube Numbering Schemes,
The Tube Store: Vacuum Tube Basics,