Google is respected in the web development community for writing simple, usable code and then making it available for re-use by everyone. mod_pagespeed is just an example of this; it's a plugin module for the Apache web server, and is supposed to speed up page delivery by optimising and caching various elements. You can read more about it at at the official Google Wemaster Central blog: Make your websites run faster, automatically - try mod_pagespeed for Apache.
So, what's the problem? Well, one simple thing to be aware of is that when you change a file (a CSS file for example), mod_pagespeed's caching may stop you seeing the change immediately. This isn't a bug, but it's definitely something to be aware of. A worse problem is that the module appears to change links on webpages to make them relative (and thus shorter). In my case, this broke a link that people use when reporting a problem, which meant that they couldn't report a problem (including the broken link -- the irony!) The link in my HTML source code was .contact, but it was being output as contact. Changing the link to https://swinny.net/contact didn't make any difference; mod_pagespeed decided that contactwas sufficient.