I assume you use the default settings. If you follow the tutorial (lesson 2), you equip your weapon (bag icon) and draw it (sword icon). The tutorial doesn't explain how you use the weapon indeed. Make sure you save your game before you start the fight with the rat.
There are two components: targetting and hitting. Ideally the rat is in the centre of the screen when you swing your weapon. You achieve this by turning your character to look in the direction of the rat, and then you adjust the height appropriately using the controls for vertical adjustments (keyboard-dependant, look up your configuration using ESC-Controls, possibly PageUp/Down or Ins/Del). Swing your weapon by keeping the right mouse button pressed (swinging in different directions has different effects).
You may also experiment with other configurations, with mouse view, full screen, etc., but I'd say do this later, get used to the standard interface first. But it's up to you.