The purpose of the blow off valve is to vent pressure in the turbo to throttle body hoses and intercooler when the throttle body is closed. If you do not have a blow off valve, when you close the throttle body, the turbo is still spinning and pushing air. With the engine only consuming a tiny bit of this air, the pressure builds up in the intake hoses and slows down the compressor wheel. This is also said to be bad for the bearings. I believe this is known as compressor surge. Venting this air allows the turbo to stay spooled and decreases wear on the turbo.
Some times this is an option, other times it is a requirement. It depends if you have a MAF and where your MAF is located. If your MAF comes before the BOV, then you will need to recirculate the air back into the system, after the MAF. There are 2 reason for this. Under some circumstances, the BOV can be open while driving. This could allow unmetered air to go in through the BOV and not go through the MAF. This would cause a lean condition. The second reason is when the BOV vents boost pressure, that air has already been metered through the MAF. Having it metered but not used would cause a rich condition. The whole point I am making here, is if you vent to the atmosphere, the MAF reading would no longer be accurate.
Other reasons to recirculate:
Aside from issues with the MAF, there are other reasons to recirculate. It is said that pushing the air back into the intake (before the turbo) helps keep the turbo spooled up. Most of the time that the BOV is being activated, it is because you are shifting to the next gear. You want the turbo to be ready when you get into the next gear. Another reason would be that since the BOV can pull in air, the air from the intake would already be filtered by the air filter. Recirculating is also much quieter than venting to the atmosphere. This is preferred in cars that aren't designed to be sporty and show off the fact that you have a turbo.
Venting to the atmosphere:
Venting to the atmosphere is not commonly done in production vehicles, but is very popular for aftermarket and modified turbo systems. Venting to the atmosphere is much simpler as you can leave the BOV open, or put a filter on it directly. To recirculate, you would need a pre-turbo hose with a port for the recirculation, a BOV that is made to output to a hose, and a hose to go between them. This would need to be designed vehicle / engine specific and would be difficult on custom, or modified systems using generic parts. You also cannot do this if you put a cone filter directly on the turbo, or if your BOV is not designed to be connected to a hose. Some BOVs have more of a horn design. This brings me to my next point, noise. A lot of people with modified turbo systems love the sound of the BOV. Venting to the atmosphere is louder.
I am trying to remember the circumstance in which the BOV can be open and allow air to be sucked in. The BOV has 2 ports - a pre TB port and a post TB port. The pre TB port is usually on the base, or it will be labeled as boost only. When the pre TB port measures X psi higher than the post TB port, the BOV opens. Many aftermarket BOVs are adjustable. Weaker settings allow it to open faster, but it may not be able to hold as much boost. Stronger setting are the opposite, more boost, slower opening. If the difference required to open is less than the idle vacuum, it will be open at idle.