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There's 2 ways of doing this:
1 - the old method: Clone the template blocks and edit these from the Admin end.
2 - Upload the template you want to change to the themes area, for usage with template-override, which is a standard feature from 2.0.14 upwards.
Example - modules/system/templates/system_userinfo.html would be placed at:
your-site-name/theme/themename/modules/system/templates/system_userinfo.html