When testing and debugging games, it is useful to see the output and errors the game creates. When running in the studio the output window shows all of these messages. When in an online server, these messages can be accessed using the developer console.
There are multiple ways to open the developer console:
The console records messages, even if you are not logged into your game. When you enter your game and open the console, you will see messages from before you joined. This can be useful when someone notices a bug in your game but aren't there the moment it happens. Each message has a timestamp localized to your timezone. Note that there is a limit to how many messages are stored.
Roblox games run two types of scripts, local scripts and server scripts. Local scripts run on client machines (the computer the player is using), and any output of local scripts will show in the local console. Server scripts run on Roblox's servers. Output from server scripts in your game will show in the server console.
Anyone running a game can see local messages (messages produced on the client machine). Only the creator of a game can see the server console.
Pressing the gear icon in the console will show additional options, namely filters. By default the console shows all messages, but if you want to toggle whether types of messages are visible or not, click the corresponding color:
In the server console, you have access to an area that allows you to run arbitrary Lua code in your game. Since it is in the server console, you can only access it if you are the owner of the place. This command bar has the same identity of Scripts and LocalScripts, which means it is not the same as the command bar in Studio and cannot run protected functions. This command bar works even if loadstring is disabled.