The Maltese Falcon is the Hub's local bar. The business used to be legit but when the former owner refused to sell to Decker they disappeared. Now the bar is a cover for his operations in the area.
In the Falcon, the Vault Dweller has several choices. They can do favours for Decker or they can gather evidence for the police. Decker's quests are both assasination missions in the Hub. Helping the cops leads to a large gunfight in his hideout under the bar. This is another instance, like Junktown, where you can be the cold blooded killer or the goodie two shoes.
Tips
The most profitable way to exploit this situation is to play both sides against the middle. To begin, you do Decker's two assasinations which are quite simple with an NPC and a half decent weapon. Then you turn Decker into the police (must pass speech check) and wipe out his gang. Fighting Decker is much harder because his right hand man, Kane uses brass knuckles to deadly efficiency. So kill him first ! Your end bonus is several thousand bottle caps and a lot of experience points.
Appearances in games
The Maltese Falcon itself appears only in the first Fallout, although it also appears in Fallout 2 as the Cafe of Broken Dreams special encounter.