To swing a weapon, you first need to go into Aim Mode by clicking the middle mouse button (or TAB button). You exit Aim Mode the same way. You may interact with objects and characters in the world while in Aim Mode by placing the reticle over them, at which point, the reticle will change into the appropriate type of cursor.
Once you are in Aim Mode, left-clicking will allow you to attack by using two-handed weapons or one-handed weapons that you hold in your right hand. If you have two weapons, right-clicking will allow you to attack by using the weapon in your left hand