What is this number and how does it effect the transactions? Why does the wallet give us the ability to change it and in what circumstances can a different number help?
The Minimum Weight Magnitude (MWM) is the amount of PoW you have to do per transaction, in a way similar to bitcoin difficulty. Specifically, this is the number of zero trits at the end of a transaction required for it to be valid. This is also why transaction hashes always end in a bunch of 9's -- because a 9 is 000 in trits.
The MWM is currently 14, but has been 15 and 18 in the past. A lower MWM means that your transactions need less time to send (this is especially beneficial on devices with weak CPUs such as phones), but the network is less safe overall, as an attacker will need less hashrate to overtake the network.
The wallet most likely provides the means to change this value for developers and other power users to test different MWMs without needing to changing the source and recompiling. It would probably not be relevant to the average user. Any transaction with a lower MWM than the network threshold will be rejected by the network, and any higher would be a waste of CPU, so it would most likely only be used for testing.