The entirety of the game happens in a small frontier town, and the player takes the role of the town's Sheriff. The scene is the town's main street, with a view from behind the Sheriff's back (the view is so close that only the Sheriff's waist, right arm and gun are visible). Various other characters appear on the main street in front of the Sheriff.
The actual gameplay mostly concerns the Sheriff discussing with the various characters via a selection menu similar to those in contemporary graphical adventures. For each line the other character says, the game offers a selection of four different responses, and the discussion progresses depending on the chosen response. Law of the West marks the first use of this now-common interaction style.
Source: Wikipedia, "Law of the West", available under the CC-BY-SA License.
In addition to wiki's description: For most people you encounter you could steer the conversation towards a peaceful end, or a voilent one. Depending on the answers you would give, characters would help you with useful tips end info, often resulting in "who you meet next". Or they could just walk away, happy, neutral, bored or angry. Or in worst case they would suddenly draw a gun in which case you would have to draw your gun in self defense and shoot your opponent before they killed you. For some characters (like loose bandits) a peaceful result was hard to achieve. It was possible to just gun down characters, but that would have disastrous effects on your score. Score was measured by the number of succesful encounters and the number of beaten villains.