To fight a monster or character in the game just point your mouse at your opponent and select the 'attack' option. Not every character in the game can be attacked, so if no attack option is visible on the menu, then the character is probably part of some quest or shop, and is not designed to be fought.
When you point the mouse at your opponent their relative strength will be indicated next to their name. In the example here the rat can be seen to be a level-8 monster. This is also coloured coded to help you decide if you should fight or not. It your opponent is weaker than you the indicator is shown in green, for equal opponents it is shown in yellow, whereas for stronger opponents it is shown in red.
Be wary about attacking monsters which are marked as much stronger than you. You could easily die before you get a chance to retreat. Some of the more aggressive monsters - particularly ones which dwell further away from the cities, or in the dungeon, will attack you if you just move near to them so be careful where you go to start with!
During the fight
Once you are fighting a monster an indicator will be displayed at the top left of the screen which you can use to select your combat style. Choosing 'accurate' will make you more likely to hit your opponent. Choosing 'aggressive' means that when you do hit your opponent you are likely to do more damage. 'Defensive' makes your opponent less likely to hit you. Whereas 'controlled' simply improves all three by a smaller amount. The combat style you select also affects which stats you gain experience points in when you kill the monster.
The strength of each combat participant is indicate by a horizontal bar above their head. If your bar is entirely green you are not hurt, it will turn red as you are hit. If the bar goes entirely red your character will die. A number will also briefly appear over you whenever you are hit by your opponent, indicating how many points of damage you have just sustained.
If you find you are losing the battle, then you can run away by clicking on the ground. You can't run away during the first few seconds of combat, but after that you are free to retreat. If you do retreat it's best to keep moving until you are some distance from your opponent as they may well pursue you.
Dying isn't much fun, so we recommend you avoid it. However most players will sooner or later get into a fight they can't win and end up losing all their hitpoints. Luckily death in RuneScape isn't permanent and after a few moments you will reappear in Lumbridge.
You keep all your stats and skills, BUT you drop the items you are carrying! It is unlikely you will be able to retrieve them so we recommend you keep as many of your valuables in the bank as possible. If you haven't attacked any other players recently (see section on player killing) you will also keep your 3 most valuable items. But you won't keep stackable items like coins, runes, arrows, etc...