Ok, how do I fix this problem with Ubuntu Software?

It means that the applicaiton you want to install requires libraries – code – from sources that Ubuntu doesn’t verify.

Imagine you have a car, and you want to upgrade the exhaust system. That might involve the use of parts that are not authorised by the manufacturer of the car. It’s the same principle.

When Software Centre gives you this message, is there any option to go ahead and install all the packages? If there is, and you’re sure you want to, click “yes” or somesuch, and the install will go ahead.

Generally speaking, it’s unlikely to cause a problem if you install the 3rd party libraries. However, Ubuntu is not responsible for maintaining that software, which means that any security issues that arise will be resolved by the producer of the code. This might mean that it never gets updated. Moreover, you might find that you run into compatibility problems down the line some time.

What software do you want to install, if you don’t mind me asking?

It pretty much means that you are installing stuff from somewhere other than the official repositories. Canonical does a good job of making sure everything it puts into the repositories works and won’t break your system, but they can’t and don’t make any such assurances about software that they haven’t checked out, so they pop that warning up just to let you know that they don’t want to be blamed for you getting screwed up by some piece of software they had nothing to do with.

I get similar errors in Windows pretty much any time I try to install anything not made by Microsoft and they are generally no big deal, though as @the100thmonkey says, it may cause issues down the road.

Have you enabled other software sources than the default ones? Otherwise you shouldn’t get that message, but if you did, it’s mostly no harm. You can, however, authenticate the sources (this doesn’t mean Canonical checks them, it just allows you to verify that the person that uploads software to those sources is the one he claims to be). If you can tell us what software sources you enabled yourself we could help you authenticate them, in most cases.