Attacks deal damage, are affected by type weakness/resistances (I'm not sure if STAB is factored in the damage formula) and may inclict status conditions.
Each Pokemon has two moves. The first is a basic attack and the second is a special (more powerful) attack.
Battles work like this:
Tap the screen to attack with your Pokemon's basic attack;
Swipe left or right to dodge before your opponent's attack hits;
Once you've filled up some of the blue meter below your Pokemon's health bar by attacking normally, you can unleash a more powerful attack by holding down on the screen before releasing.
Here is an image showing the blue bar:
So if you have a special attack with 3 blue bars (you can check this in the pokemon status screen, where you can also see the listing of attacks it can perform) you would need to attack with the basic attack to fill up 3 blue bars (the ones below the green health bar in the image).
Sources: personal experience and this wiki page