Any Figure (A figure is a creature or building) that has an Attack Value is able to attack the enemy's figures. There are two types of attacks, Melee attacks and Ranged attacks. Attacking cause the figure to dim. So a figure can only attack once a round and can't move after it has attacked.

Melee

This is the most common way to damage your enemy's Figures. In order to make a melee attack on a figure you must be able to move into the target's space if the target wasn't there.

For example here the Axe Captain can melee the Elven Scout since he is close enough. But he can't melee the Ulg Turtle since the Turtle is in water and he can't melee the Thistle Falcon because it is Aloft and he can't melee the Dellenari Beastmaster because she is too far away.

The exception to this is when the enemy attacks you. In which case you can Battle Back.

Shooting

Some Figures have range. A Figure with Range can shoot any enemy figure within that number of spaces. A Figure can't move and shoot. It must choose to do one or the other each round.

Earthworks is an example of a Figure that can shoot. It can target any figure within 2 spaces.

Damage

Figures that have less than 1 Health don't die until the end of the round. This gives them a chance to retaliate if they were attacked or do some other last dieing action.

You can see that the Elven Scout has been attacked by the Crypt Doctor and now has 0 Health. It can still take an acction this round but it will die at the end of the round unless it is healed somehow.