Most of the information you need for the Fight button, is within the Combat section of the Wiki as it will give you a break down of how everything works.
This page will give you just a quick run through of what you'll see and how you actually fight Monsters.
In the top left, you'll see the Locations. If you are are just starting out, you'll want to stick to the Wasteland. Each button represents a location.
The current locations are:
Underneath the locations, you'll see the Monsters list. Clicking on a Monster will highlight it in blue, and then show you it's respective stats, drops etc.
To fight the monster, you then need to click the green button which states “Fight NAMED MONSTER”. You will then begin to fight that particular monster.
You can change monsters at any time.
On right hand side, you'll see information about your Last Fight.
This is basically just a log of what happened.
It'll show you how many monsters you killed (or if you died), how much health you had left (same for the monster) and what you found.
The last section shows the log of the fight, so you can see the damange counters between you and the monster that you fought.