What do the numbers on stick electrodes mean?
Those are classification codes, the most common in the US being the American Welding Society (AWS) numbering system. You may see labels such as AWSE6011, E6011, or simply 6011. These values all mean the same thing, some just provide a bit of extra information.
Carbon Steel
{AWS}E6011
The first part simply tells you which standard is used. in this example, it's the American Welding Society numbering system. This value may or may not appear printed on the consumable.
AWS{E}6011
The E in this example, represents the consumable type.  E represents an electrode, which means an electrical current causes the filler material to be deposited. This value may or may not appear printed on the consumable.
Consumable Types
E - Electrode.
R - Rod.
ER - Electrode or Rod.
EC - Composite Electrode.
B - Brazing.
RB - Electrode, Rod, or Both.
RG - Oxygen.
IN - Insert.
EW - Tungsten Electrode.
F - Flux.
AWSE{60}11
The first two (or 3) numerical digits represent the tensile strength of the filler material, measured in thousands of pounds per square inch. In this example, it would be 60,000 psi.
AWSE60{1}1
The next numerical digit, represents the position the consumable can be used in.
1 - All positions (Flat, Horizontal Fillet, Vertical Up, Vertical Down, Overhead).
2 - Flat and Horizontal Fillet only.
3 - Flat only (deprecated).
4 - Flat, Horizontal Fillet, Vertical Down, Overhead.
AWSE601{1}
The last numerical digit tells you what the coating material is, and which types of current can be used.
0 - High Cellulose Sodium - (DC+)
1 - High Cellulose Potassium - (AC, DC+, DC-)
2 - High Titania Sodium - (AC, DC-)
3 - High Titania Potassium - (AC, DC+, DC-)
4 - Iron Powder, Titania - (AC, DC+, DC-)
5 - Low Hydrogen Sodium - (DC+)
6 - Low Hydrogen Potassium - (AC, DC+)
7 - High Iron Oxide, Iron Powder - (AC, DC+, DC-)
8 - Low Hydrogen Potassium, Iron Powder - (AC, DC+, DC-)

Stainless steel
{E}308L-17
The E here is the same as with carbon steel, see above.
E{308}L-17
The numerical value here represents the filler metal that the consumable is made of, and will be some grade of stainless steel. Note that the filler material, doesn't necessarily have to match the material being welded.
E308{L}-17
This character may or may not be present, and indicates the percentage of carbon in the filler material. If the character is present, it can be either L for low carbon, or H for high carbon.
E308L-{17}
The digits after the dash, represent the coating material and allowable current type.
15 - Lime based coating - (DC+)
16 - Titania or Rutile - (AC, DC+, DC-)
17 - Silica, Titania - (AC, DC-, DC+)