Roblox games can be saved online on the Roblox servers or offline on your local machine.
The recommended method of saving progress on a game in Roblox is to publish it. Publishing a game pushes a save of the game to Roblox's servers. Publishing in this way allows the game to be edited from the Roblox website on any computer that can run Studio. Publishing also allows you to activate the game on the Roblox site which enables others to play your game with the Roblox client.
To publish a game select "File > Publish to Roblox As...". This will open a dialog where you can select which slot the save goes to. You can either select a new slot or overwrite an existing save. There is no limit to the number of saves you can have on Roblox. If you select a new slot by clicking on "New Place" you will be prompted to provide a name and description for your game.
If you are in a Roblox Group, you also have the option to publish to a group slot through this dialog. This operates the same way as publishing to your own slot, but keep in mind other members of your group can save to group slots as well. If you are overwriting an existing slot, it is recommended to coordinate with your group so you do not overwrite another member's work. See Group Games for more information.
If you have already published your game in a session of Studio, or if you have opened the game via the Edit button on the Roblox website, you can select "File > Publish to Roblox" to save the place to the same slot as it was already saved to. This does not open a new dialog window.
You can also save your place locally to your computer by selecting "File> Save As...". This will prompt you to provide a name for the place as well as a local destination. This saves a file in either an .rbxl or .rbxlx format. If you have already saved your game in a session of Studio, or if you opened a local file, you can select "File > Save" to quickly save your changes to the same file.