What do the numbers after skills indicate?

To elaborate on GinWolfe's answer:

The number doesn't indicate the damage of a skill. If that were the case, skills with 40+ would instantly defeat most opponents. Instead, this number indicates it's power level, which direct influences damage.

All pokemon games use the same principle of power, although I cannot confirm that it follows the exact same mechanics for calculating damage. For more information on power in previous pokemon games, check http://bulbapedia.bulbagarden.net/wiki/Power.

As GinWolfe already explained, the bars indicate when the second (special) move can be activated. Stronger moves generally have longer bars, requiring the player to charge it up (by attacking with the pokemon's normal move, like Tackle or Quick Attack) for longer period of time before it can be used. Multiple smaller bars simply means that once the first bar is filled, the second bar starts to fill so you can use your special skill multiple times in succession for each filled bar. These special moves can be activated by either pressing and holding on the enemy or by tapping one of the filled bars in the top left.

Source: personal experience

I actualy think the numbers after each skill are actualy the attack speed because no matter how much your CP (combat points) is powered up it never changes like what it is supposed to. And the basic also have a number after it so i tought it is speed. If your combat points is higher your pokemon should hit harder but those number dont change. if your pokemon il 100cp and the other one 300cp they do have the same numbers after each skill(they must have the same skills)

Thats just a superstition but i'll let you guys judge it and tell if u have so news about those mysterious numbers.

The basic attack number I do not know what that means anymore than anybody else. The blue bars are how many you have to fully fill up before being able to use the special attack. However, the number beside it does not do that number in hp damage. For example, my Vaperon has a special attack of 45. If it truly did 45 damage every time I used it, I would wipe out even high CP Pokemon in one or two blows from my special attack and that's not the case. In fact, I haven't read this anywhere but I have won over 100 gym battles and I can tell you that using my special attack causes a my own health to take a major hit. It's actually better to use the normal attack to keep the opponent from being able to power up their special attack and then only use yours when they're weak enough to cause them to lose by using it. I was using my special attack as soon as it was charged and the Vaoeron that I spoke of is a 1611 CP and wasnt losing hardly any health from the weaker opponent but list significant health from using its own special attack. It may just be something I'm doing wrong, but using special attacks definitely hurts your own health and does nowhere near the damage as the number beside it.

The number listed next to a move is its power. On GamePress' webpage on Damage Mechanics, it fully explains the formula to express how much damage a pokemon will deliver. A move's power is only one of five variables in the formula.

Special attacks require energy and must be charged before use. The blue bars indicate how many special attacks per 100 energy units are possible. More bars generally correspond to moves with less power but can be charged faster.

Every move also has a duration or how long it takes to complete. Hence, comparing moves' relative strength cannot simply look at their relative power. Several websites can tell you every move's damage per second (DPS) to help you compare moves. However, they do not factor in type effectiveness nor same type attack bonus (STAB).

Total damage for the second Special Attack appears to be the number shown x number of blue bars. Air Cutter has 4 bars, so total damage is 100(4x25). Twister has 5 bars, so total of 75(15x5)

I think the number in the bar indicates how many times you hit the screen to fill up that bar to use the special attack and that's it.