Discord by default will show what game you're playing because it keeps a database of common game executables and their corresponding games (i.e., if there's a program running on your computer called "LeagueOfLegends.exe", Discord knows to write "Playing League of Legends" as your status). However, if you've ever played a more obscure game, you may notice that Discord does not automatically update your status at all. This is because Discord does not recognize the game's executable, or it doesn't have the game title in the database.
You are able to tell Discord which programs are "games" and what those game names are. The process is detailed on their blog. Basically, while a program is running, you can select it from within Discord and give the executable a name. Then, whenever that program is running, Discord will update your status message.
This only works with games that are not already verified. Most popular games are already recognized, so you won't be able to customize a League of Legends message. However, it sounds like you want a custom message for when you're not playing games. If that's the case, there's a kind of hack to work around that - just add Notepad (the windows text editor) or some other lightweight program to Discord's list of "games", then edit it's name to your status. Whenever you want to display that status, just open Notepad.