Bitstamp is one of the longer running Bitcoin exchanges. It has been around since 2011 and is a licensed exchange with the Luxembourg's Ministry of Finance
It is a good option for traders and those buying large amounts of bitcoins.
Verification
Bitstamp requires extensive verification including the following information:
Full name
Date of birth
Country of residence
Phone number
Address verification
ID number and scan of photo ID document
Fees
Because Bitstamp is geared towards traders, it also has confusing fees if you use the exchange. The fees you pay depend on your total volume. Unless you trade high volumes, you will likely pay 0.25% per buy.
If you buy bitcoins on Bitstamp with your credit card then the fees will be 8% for purchases up to $500 or €500. There are other exchanges that offer lower fees for buying bitcoins with a credit card or debit card.