That is a protoboard/veroboard/perf board/etc.
Does a board like that require soldering? Is this at all similar to a "breadboard?"
Mostly, yes, you would solder it. You could also use wirewrap, but frankly, that kinda defeats the purpose.
It is similar to a breadboard, except that it has a different layout, and is used for (mostly) permanent or final circuits. Breadboards allow you to move things around easily, because they are solder free.
Is it possible to use a board like that to connect with GPIO ports?
Off the bat? Maybe. But with a few caveats. The layout of the one you pictured wouldn't be ideal for a header used on the RPI. You could solder it on directly, but that would make removing it hard, you most likely would need to wire around it, you would have to put it on upside down, etc. You could use a female header to get around some of the issues. Most of the time, people use a cable between the RPI gpio port and a circuit board.
See http://lookmanowire.blogspot.com/2012/07/raspberry-pi-and-xbee.html or http://www.atbrask.dk/?p=27 for examples on a perf board connected directly to a RPI (with a header)
If not, what would someone use a board like this for?
Practically most hobbyist circuits can go on a perf board like that.
And you can google image search "raspberry pi perf board" for even more examples.