Amazon's Kindle Direct Publishing allows you to create an author account where you publish and manage your e-books through Amazon. You have to provide your real name and social security number as well as bank account information so that they can pay you and also so they can report your earnings. However, you do not have to use your real name on any of the titles that you publish through them. They allow you to provide any name you want for the author, and you are not required to provide a publisher's name. If you want, you can tie your author account to your titles, but it is not required. I currently have over thirty titles published with them using my own name and three different pseudonyms.
I have only recently started using B&N's Nook Press, but it appears that they provide the same capability. I currently have twelve titles published through them using two different pseudonyms. My account is registered in my real name, but the books I have published through them do not have my real name tied to them.
Smashwords, on the other hand, is not nearly as friendly. If you want to keep your identity separated with them, you will need to create a separate account for each one of your pseudonyms. I currently have only my real name and one pseudonym distributed through them. I am looking at other options that I may use in the future to replace them.
As mentioned in one of the other answers, there is no hard and fast guarantee that you will be able to remain 100% anonymous, but it is possible to publish under a different name and still get paid using your real name and current bank accounts.