How do the Attack and Defense stats work in Pokémon GO?

Here is a graphic explaining the process

All attacks have a base damage stat. That's the number you see, and no matter what pokemon you have, attacks like vine whip will do the same base damage. But, you're right, there absolutely IS an equation that uses this and the pokemon's stats to determine a final damage. Since pokemon go hasn't been out for that long. The exact formula in each battle is basically impossible to know, because you don't know your opponent's defense, but in the mainstream games, it's

((2xlevel+10)/250 X (attack/defense) X Base + 2) X modifiers

Where attack is your pokemon's attack, defense is the defending pokemons defense, Base is the move's base damage (that constant, like vine whip has) and modifiers are things like STAB, types, etc.

In Pokemon Go, they've changed a lot of individual numbers around, but they most likely use a similar formula to the one above. So, essentially, base attack is not a set damage. It might SAY something like 20, but if your opponent has a great defense, it could be less, or if it's super-effective, it could do a LOT more.

formula is from here

Just in case the link breaks here's a text version

Yesterday (July 16th) there was an info dump of data-mined stats from the game's code. We can now calculate what makes each Pokemon tick and how strong they are.

Base stats

There are 3 (2 are hidden) stats that make up a Pokemon's CP (Combat Power): Staminia (HP), Attack, and Defence. These values are produced by taking the Pokemon's stats from the main games and running them through this formula:

Base Attack = 2.6 * (Atk^0.46)(SpAtk^0.46)(Sp^0.04) + 3
Base Defense = 2.6 * (Def^0.46)(SpDef^0.46)(Sp^0.04) + 3
Base Stamina = 2 * HP

Because the speed stat is weighted so lightly in this formula, Pokemon that relied on their speed in the main games (such as Electric-types) will find their stats lacking in Pokemon GO, as we see the bulky Water-types like Vaporeon and Slowbro that are normally very slow have a big stat advantage.

STABs and type effectiveness

^{This section is missing.}

IVs

If you thought the grind was over after finally getting that Vaporeon with Hydro Pump, think again. Supposedly IVs (Individual Values) are in GO. I can't confirm it however.

Attack speeds and DPs

Moves have different, set attack speeds.

Don't let the Power listed on the move fool you. Generally, moves with higher power will have slower attack speeds.

That isn't to say each move has the same DPs: Water Gun for example is one of the best DPs moves you can use, whilst Thunder-Shock has the same attack speed but deals half as much damage.

You can calculate a move's base DPs value by plugging the values in here:

DPS = (1OO0/attack speed in ms)(attack's power)(1.25 if STAB)

To find out a specific pokemons DPs value relative to others, multiply that move's base DPs by the user's Attack stat.

For example: Let's find the DPs value of Vaporeon:

Vaporeon has a base Attack stat of 186.
Vaporeon can only learn Water Gun as his basic attack.
Water Gun is 10 power and has an attack interval of 500ms. You can attack twice per second with WaterGun (1000ms/500ms).
The effective power is now 20 but Vaporeon also gets a STAB boost since it is Water-type. This brings Water Gun up to 25 effective power (1.25*20). Multiply his attack 186*25 = 4650. This puts Vaporeon's DPS at #7th high- est Combining this with the fact Vaporeon has one of the biggest health pools and a decent defence makes him the top pokemon by far at the moment. It's no coincidence you see Vaporeons at the top of many gyms.

So why does evolving my Eevee into a Vaporeon make it so much stronger than Flareon or Jolteon?
Jolteon, like most electric-types, relies on his speed. As discussed earlier, this hurts his base stats which is why you will see an Eevee that turns Vaporeon having 25-30% more CP than an Evee with the same CP that turns into Jolteon. That extra CP is an indication of the extra HP that Vaporeon is getting which is roughly double that of Jolteon's. On top of this, Jolteon's only basic attack Thunder-Shock has a base DPs literally half that of Water Gun. Theoretically if they were to fight, a Vaporeon should walk away with over half his health remaining after trashing a Jolteon!
Similarly, Flareon's Ember deals similar DPS to t-shock (11.9). Luckily for Flareon his evolution comes with extra Attack (238), which still isn't valued as highly as HP, but better than Speed. Although slightly higher attack than Arcanine, his other stats are lacking in comparison. Compared to Arcanine's much higher HP and DEF paired with his 15 DPS Bite which would do similar damage to an Ember vs Super-Effected targets, Flareon's Ember with a dps value of 2833 can barely even compete against top-tier Pokemon with fire moves as lacking as they are. The only seemingly semi-viable Fire pick is Charizard. And that's only when you use him with Wing!

So how do we counter this monster of a mon Vaporeon? Well, right now you don't.
The only pokemon that can even come close would be Venusaur. But thanks to 1.25x/0.8x effectiveness, Venusaur boasting a respectable 3808 grass dps value with a decent Defence, just barely loses to an equal Vaporeon because of Vaporeon's HP advantage. Exeggutor can also come close, but lacks the grass-type basic move to seal the deal.

Personally, I can't wait for trainer battles to come so we can have nothing but 6v6 Vaporeon battles!