Bitstamp is one of the most popular fiat to Bitcoin exchanges. It was one of the first big exchanges and it provides customers with a platform that is secure and easy to use. It was originally started in Slovenia but has moved its' operations to Luxembourg.
Bitstamp uses MultiSignature technology for its hot-wallet, as well as a fully insured cold-storage option where kept off-line. They also offer a number of other security features such as two-factor authentication and confirmation emails to enhance account security.
Bitstamp provides the option for limit orders, stop loss, and trailing stop orders. They have an easy to use API that can be integrated into user applications. T
Overall, Bitstamp is a secure and reliable exchange with good liquidity and a solid infrastructure. It is a good option for beginning traders as well as more advanced crypto holders.
Trading Fees: Tiered system depending on volume ranging from 0.1%-025%

BitFinex is a bitcoin to fiat exchange, a margin trading exchange, and a liquidity provider. The platform offers a number of features that expand the financial positions you can take - for example the ability to short Bitcoin via margin trading.
The exchange also offers a set of other cryptos apart from bitcoin such as Litecoin and Ethereum.
Deposits and withdrawals take place via the standard bank transfer means with fees of 0.1% and a minimum of $20. Bitcoin and Litecoin are free.
BitFinex takes security seriously and have moved from a hot wallet cold wallet set up to segregating customer funds where each user has access to their own wallet which they can review on the blockchain.
Bitfinex was hacked on 03 August 2016 and around 120,000 BTC were stolen.
Fee of 0.1% for most trades