Hi,

So I found a free font and emailed a document containing some of the characters to myself. When I opened it on another computer the font had changed. Can you only use it from the computer it is on?

You may need to be a bit more specific about what kind of document it is. If it's a Word document, you will need to tell MS Word to embed the font, otherwise, when you open it on another computer, without THAT font installed it will be replaced with a substitute font.
https://support.microsoft.com/en-us/kb/290952
When asking tech questions tho' it is as well to include as much info as possible.
App used to create document and version of said app, OS version of all computers involved, and the name of the font you are having issues with would be the minimum infos to get you a quick resolution to your problem IMHO. If the above doesn't sort you out, post this info and I'm sure someone will be able to find a solution for you

Yes, the font must be installed in all computers where the document is to be opened. The only exception is if the font is embedded in the document. This is not true in your case.

Not all programs can embed a font to a document so you need to check your program if it has that feature.

And not all fonts allow embedding.