You can send your layouted PCB to a professional service that will produce an industrial-quality board. These will usually require your files in the industrial standard format "Gerber" or "RS-274X", which you can export directly from Fritzing.
It is however still a bit tricky to use these services: They will require some expert knowledge, and are commonly oriented towards mass production, not the production of a single board. Some companies have recently specialized on more individual productions (so-called "batching" or "pooling" services).
Fritzing Fab is our attempt at this, making it super-easy: Just upload your Fritzing file to http://fab.fritzing.org and get a fancy black-on-white board of your project. No strange parameters, top quality made in Germany, all at a good price. And as usual, all profit goes back into the further development of Fritzing.
There are several techniques to produce a PCB on your own, requiring only a little bit of equipment. The most common one is an etching process, very similar to classic development of a photographic film. It's fun to do this, but be prepared to spend some time on perfecting the process.
How to etch a PCB by yourself.
How to cnc a PCB by yourself (external tutorial)
A quick run down of different ways to produce a PCB.
Handy pdf for testing the quality of your etching setup.