< So I was wondering what the mechanism is called that will attach to my bike >
It's called a windtrainer, or sometimes just a trainer. I have a Cateye Cyclosimulator. Pricey (350 on sale), but for me it was worth it. Factors to consider include construction quallity, stability, noise, computerized readout or not, etc... The computerized readout is particularly important to me - knowing my speed, distance, cadence somehow motivates me.
If you love biking and can afford it, invest in a pricier model. A cheapo that shakes and vibrates will annoy you (you can find less expensive trainers that are of good quality - they just have fewer options). But if you don't really like biking all that much anyway, anything you buy might end up in a closet collecting dust. Riding stationary can get boring. I've been thinking about getting a rowing machine myself, just for a change of pace.
Call a few bike shops in your area and see if they will let you come in and try out the models they sell. Make sure you tell them you are looking for a stationary trainer not rollers (another typer of trainer but probably not something you are looking for).