Disclaimer: I go with tarps or tents instead of Bivy's but have used them before.
- Protects you from water
- Protects you from strong winds
- Some ventilation (through openings or fabrics) to reduce condensation and allow fresh air flow
- Large enough to comfortably fit you and not too much bigger (the less empty space, the easier it is to stay warm inside, but you also want to make sure it fits you plus whatever gear you'd want to bring inside)
- A hole for your head that you can open or close depending on preference
- Bug netting at the opening to protect against insects
- A reliable design for keeping the shelter up off of you - particularly your head - while you sleep (it can be hard to prevent condensation altogether so keeping the shelter off of you can help keep you and your bag dry overnight at least)
- User-friendly: relatively easy to open and close, including while cold; better yet, easy to setup/pack away in case you need to move it in a cinch
- Tough bottom to handle rough terrain
There are a whole range of options from simple plastic tubes to something more like a small tent.
Ones which have no real supporting structure at all, essentially waterproof sleeping bag covers need to be more waterproof than those which are supported a they will have folds and hollows which can trap water. Similarly there is always the possibility that you can get runoff or puddles forming underneath them if your campsite is less than ideally located and so really need to be gore-tex or similar which can withstand a reasonable head of water. These have the advantage of being light and simple plus if you move around a lot when you sleep or like to sleep in an odd position they can be more comfortable, plus they allow you to sit up so you can for example, make a cup of tea in the morning before you get completely out of your sleeping bag.
A more structured one, the most elaborate of which start to overlap with small tents will tend to give better weather protection for a lighter fabric weight as they shed water better and the space between you and the skin gives a bit more insulation and ventilation. Some people find these a bit claustrophobic as being in a small tunnel can be worse than a close fitting, although this is of course very subjective. Conversely having a hoop to create some space around your head can make it more comfortable if you get stuck in one to sit out a storm.
Size is also an important consideration, especially if you are tall or heavily built, similarly you don't want to to be crushing the loft of your sleeping bag, considering also that in some circumstances you may want to sleep wearing a moderate amount of clothing. Fit and overall comfort is best assessed by actually trying on out with the sleeping bag and clothing that you intend to use. You also need to consider how much gear you want to be able to keep in it overnight, clearly most models won't accommodate a large pack.
Equally you may want to experiment with what design of opening works for you, if you expect good weather it is often more comfortable to sleep with at least some opening around your face.
Even with the best breathable fabrics and you will tend to get a bit more condensation than you would in a tent and this can be a problem for down bags even if now water gets in form the outside and it is very difficult to make a bivi bag 100% waterproof in all possible conditions.
It is also worth considering that even very basic secondary shelter greatly magnifies the performance of a bag this could be a lightweight tarp or a natural feature of your campsite.
As an aside it is also worth mentioning the Buffalo sleeping system as this inherently works well with bivi bags as it is a lot less sensitive to getting wet than most other sleeping bags although it is a not to everybody's taste and doesn't compress as well as lofted insulation.