What makes grease appropriate for bikes?
Phil Wood, the owner of the bicycle tool and components company of the same name, once commented that his number one requirement when he was buying barrels of grease for his company, was that it be the exact same green color shade as the previous batch, so that he didn't get too many phone calls asking what he'd changed.

Point is that the "bike specific" greases generally aren't. 

As far as functional necessities:


Not too thin - A thin grease, like Slick Honey, will run very smooth, but only for a very short time. If you want ideal performance, and don't mind overhauling your bike every couple weeks, use this.
Not too sticky - Motorcycle/Automotive type bearing grease tends to be very sticky and very thick. They are designed to thin out at higher levels of heat and friction than a bicycle will reach. So it will last a long time, but will feel tight, and add friction to your bearings.


Generally, buy a tube of Phil Wood or Park tool grease, and feel free to use anything with a similar purity and consistency in the future.

Edit: After reading Matthew's answer on this page, I realized I hadn't pointed out the 2 cases where the grease matters. In the case of Carbon, see here. In the case of Titanium, with any non carbon frame, you must use a copper paste grease, commonly sold as TiPrep. it is messy, annoying, gets all over everything, but it stays put, which no other grease i am aware of does for Ti.