Owning a monopoly is when you own all properties of a given color on the board. Properties come in groups of two or three. You must own them all at the same time with one player to unlock this achievement. You can either hope to luckily roll to land on them to be able to purchase the properties, or just trade for them from whatever other player owns them. This is easy in pass-and-play. To initiate a trade, hit the Trade icon (has two arrows, right next to dice roll or end turn icon). Then select the player you want to trade with and just swap for the properties you need.