What is a fountain pen? It is a pen that contains it's own reservoir of ink, which flows through a the end of the pen, called the nib. This sounds exactly like any other pen, but the ink flows freer than your normal ballpoint pen, and the pen itself looks approximately 117% fancier.
See, way cooler than a Bic
Like I was saying, the ink flows from a reservoir, either a cartridge one or from a bladder you manually fill within the pen itself. The bladder pens are the cheaper way to go, so that is what we will focus on the rest of the way.
There are many different brands, with some luxury brands (Mont Blanc) being incredibly expensive, and in my opinion the height of stupidity...who the hell is dropping several thousand dollars on a pen?!?!?!?!? That is slap across the face worthy levels of wasteful. A quick browse on amazon shows many in the $12-$25 range, which is perfectly fine. Personally, I have a Pilot MR, which I like a lot.
Usually the pen comes with ink already in it, but when the time comes to refill, I recommend Noodler's Ink. It is cheap, comes in a huge variety of colors, and is of very high quality. (http://noodlersink.com/ is their website, but you can buy their stuff straight from amazon)
The initial cash outlay for the pen and ink can get up there in price (~$50), but over time it is much cheaper than disposable pens, has a much lower environmental impact, and just looks way classier and makes even note taking feel special.