There are a lot of online services where you can keep your bitcoins. You just need to choose what is more convenient for you, depending on what features do you need.
All Bitcoin Exchanges can hold bitcoins and some of them allow you to instantaneously transfer bitcoins between users, which can be a big plus if you want to accept payments.
Other popular ewallets include instawallet, walletbit and flexcoin. Flexcoin acts more like a real bank because it pays "discounts" (interest) to users. Tradehill is also preparing an ewallet at bitcoin.com. blockchain.info offers an ewallet with the unusual feature that they don't have access to your private keys.
As for security, it is hard to know what is going on from the outside. Some exchanges keep part of their coins offline, which is good. Flexcoin is also planning to introduce a cold storage feature.
New services appear every day, so it is hard to point out a single big "bitcoin bank" right now. Check a few and choose one that suits your needs.