Fighting is generally accomplished by clicking on an enemy, which attacks it with the weapon you've got equipped. When it's killed, you receive experience. If you are killed, the game ends.
Often, you will want to use a special ability, also known as a skill, on an enemy. See THE RADIAL MENU for more on this.
EXPLANATION of TURNS
In this hypothetical situation there are 9 tiles arranged like this:
1 2 3
4 5 6
7 8 9
Tile 5 contains a PLAYER. Tiles 3 and 9 contain monsters.
PLAYER moves from tile 5 to tile 2.
Monsters seem to move AS the player moves visually, but they really make the decision a split second after the player moves. So monster @ 9 moves up to 6 to get closer to the PLAYER as the player moves, but Monster 3 does not move, because he is now within attack range of PLAYER. So the monster's turn is spent attacking. He attacks, and any other monsters (possibly ones outside the grid with ranged abilities) would go as well.
SPECIAL CASE: Attack of Opportunity.
In a special case, Monsters have an opportunity to get an extra bonus attack on the player, attacking him AFTER the player has ISSUED the command to move, but before his character gets to actually move. This is called an Attack of Opportunity in D&D. This attack is a free extra attack and doesn't count as the monster's turn.
This happens when a player is adjacent to an enemy, and moves off axis with the enemy - meaning if the enemy is to your south or north and you move east or west, or if the enemy is to your east or west and you move north or south.
1 2 3
4 5 6
7 8 9
Player is at tile 5, monster at tile 6.
If the player moves DOWN to tile 8 - right when the player issues the command, before he moves, the monster adjacent to him will attack him. Then after that the player will move, and the monster won't move, because they already took their turn attacking.
Notes: The major reasoning for having these AoOs is for when there is an enemy chasing you who is constantly 1 tile away from you. POWDER, if you notice, also does have this system, in that as long as you're running a straight line he wont hit you, but once you make a turn he'll be able to swing at you. Same deal here. This is so that you can't just perpetually run away from monsters, and also it's just a solid tactical element for combat in general, hence its inclusion in D&D.
The player does not get AoOs on enemies.