I created a model using ModelBuilder in ArcMap.
How can I share this model as a toolbox to run it in another computer?
The tool is expected to be shown under system toolboxes and users can run it as existing tools.
There are a few steps you need to take to ensure that your model will run on another computer. Some of which include:
Check out the following ESRI help page for some details: Sharing Custom Tools
The .tbx file is portable. So you can copy it into a network file store, or usb to allow access to it, or send it as an email attachment (if it's not too big) to be saved in a location that ArcCatalog can get to. As long as the other ArcCatalog user has the proper licensing, and the tool has been saved to a folder connection that the user has listed in ArcCatalog, the tool will be recognized.
If there are dependencies such as files, folders, scratch space then those dependencies need to be configured on the "new" machine that the tool is being run on.