Alpaca is a mount added to Final Fantasy XIV DLC Dawntrail. To unlock this fluffy creature, you will not have to do anything complicated, but still you will need a massive amount of time. It will become available after finishing Dawntrail Main Scenario Quests. When you complete the last mission, you will get Alpaca Whistle which allows you to call your new mount.
However, you can test Alpaca earlier. During main quest line, on level 90 you will be able to complete Mablu’s Dream quest. This is when Aplaca is your temporary mount.