After having found your lovely apartment, you have to (you should at least) register as a tenant. Thanks to this registration, you are now able to insure your place, open a bank account.. and pay taxes (it would not be funny otherwise, don't you think?).
You could actually live (illegaly) without this BSN (Burger Service Number) but you would not be able to do most of the things I said earlier and there are also some chances your employer cannot pay you. Too bad, isn't it? Moreover, you being registered might get you some governement help to pay your rent.
I already told you, without BSN, no bank account but you should know that VISA card are not accepted in all shops (the two supermarkets I shop do not for example). You can always withdraw money from an ATM but your foreign bank can charge you because you withdraw from ATM of another bank in another country. In addition, public transports company (GVB, Connexxion, EBS) ask you to pay your abonnement on the Internet via iDEAL.. which is a service only offered by Dutch banks.
To obtain your BSN, you have to take an appointment at the Town Hall (Gemeente in Dutch). You have two options: either you phone them (+31 (0)20 624 1111) or you visit them (Amstel 1, 1011 PN).
Be quick! Like in every public service, you can wait a lot before getting your appointment (at least two weeks!). You'll need your (valid) ID card or passeport, a birth certificate, a rental contract with your landlord, your work contract (if you have one) and a copy of your landlord's ID card if you're his/her first tenant.
Spare an hour and a half of your precious time for this appointment.
You're now a citizen of Amsterdam, how do you feel?
Sources:trips, amsterdam, bsn, burger service number, register, registration