Snowboard bindings, believe it or not, can actually be a more important decision than the board when buying your new snowboard setup. The mobility of your bindings affects the mobility of your board. They must also have the proper size, flex, and type according to your preference. As far as binding company's go there are a lot of them that make snowboards as well as bindings. There are actually only two company's that actually specialize in bindings alone (Flux-Bindings & Union Bindings). If you end up buying a snowboard made by Flow, Salomon, Forum, Ride, Nitro, Burton, Rome, as well as some others it's probably smart to go with bindings by that company as well. They design their bindings to go with their boards. You can usually find more info on which bindings go better with which boards on their websites. However Flux and Union bindings are never a bad choice considering the fact that that's what they specialize in.
There are many different types of bindings for different types of riding. Some of the different bindings range a lot in price based on whether they are more of a high or low end binding from that particular company. I would highly recommend buying a more expensive pair just to be safe. I've had lots of friends who've injured themselves because their bindings have broken mid air or even just while riding down the hill due to defects (there isn't much you can do to prevent yourself from sustaining an injury no matter how experienced you are).
Most bindings are pretty similar in most areas, some are different as how you get in and out of them. There are also many different types of toe straps. They're all designed for different things--some are designed to be light weight, some are designed to fit around your toes better. Again this is all based on your personal preference. The Flow bindings are the ones that allow you to put your feet in from the back. There are no toe or ankle straps, they just have one piece that fits on the top off your boot with 4 points of pressure.
As far as the flex goes I would recommend a more flexible binding for more mobility. If you are a more experienced rider or you are more into powder riding or half-pipe riding a more stiff binding is probably better. Another thing to sometimes watch out for that will also effect your flex and mobility is the heel cups (that's the horseshoe shaped piece that's in the back of your binding that your heel leans against). Heel cups are most commonly made of either plastic or aluminum. It's better too stay away from the aluminum ones only because they tend to put more pressure on your snowboard underneath the binding causing the snowboard to crack or break. Plastic in my opinion is a better way to go and tend to be more flexible as well (you cannot choose a binding and then get the option of aluminum or plastic, they come how you see them).
The last thing about your bindings is setting them up properly. After you've picked out the board and bindings that you feel is best for you make sure that they are set up properly (this may take a few runs before you can get a feel for how you want everything adjusted). On each binding there are heel and toe pads or sometimes referred to as "gas pedals". They need to be adjusted so that they reach as far to the edge of your board as they can without hanging off. This allows you to have more response on your heel and toe edge. If you are buying your first snowboard or don't know how to properly put it together, your local snow shop or snowboard tech can help you adjust your board and bindings for you.
Try to stay away from Aluminum hell cups. They can put added stress on the board when you move pressure from heel to toe and lead to damage.