The *.map file is a source map. It contains informations like function positions of the non-minified source code of Qtip2. Notice that I used the minified version of it on my webpage.
Source maps help to develop with a postprocessed (e.g. minified) source code without losing the ability to look into the code, for example when errors occur.
The chrome debugger as you can see on the image in my question above downloads it by default. After disabling that function in the debugger settings the file won't be downloaded any longer.
Take a look at this video for a short information or this documentation from Google.
Firefox offers the same feature as you can see here.