Play "Digimon Masters" online by downloading the game client from Joymax's official website. If you're under 13, make sure you have a parent's permission before you download anything. From the main landing page, select the option to go to the "Digimon Masters" site. From the DMO page, you need to create an account to play on the game's servers. After setting up an account, download the client and you are ready to play.
Once you are in the game, you are prompted to select your name and avatar. You can customize your avatar however you see fit. Then, you can select a Partner Digimon that you use to fight and accomplish quest objectives. The battle system is played in real time, like many other massively multi-player online role-playing games such as World of Warcraft or Guildwars. It's up to the player to decide what he wants to do from there.
While the game is free to play, it is supported by the purchase of micro-transactions that allow you to buy Digimon, inventory expansions, and other enhancements that may make the game easier or more enjoyable. Micro-transactions are completely optional, and it won't hinder your gaming experience to play without them.