As a web developer, I'll address your concern.
Thanks to all the FUD put out about cookies by the media, most people are now paranoid about them. That has made our job that much harder.
There are two ways we can preserve state across pages on a site. By that I mean how we know that it's you asking for a page as distinguished from the other million visitors using the site at the same time. The two ways are: cookies, and session ids. Both do essentially the same thing. but in different ways.
A cookie stores the state id on your computer, while a session id stores the state id in the URL if using a GET method (you've probably seen these; they look like a mess of numbers and letters, like "www.mysite.com/do.php?sid=a5011C6&p=5 etc... the "a5011C6" bit is the session id) or passed as a parameter if using a POST method.
The main difference between them is that cookies preserve state even if you leave the website - the cookie can be used to remember that you logged in and to restore that logged-in state if you return before the cookie expires. Session ids, on the other hand, lose state when you leave the site (because other sites obviously won't carry your session id), so with them you have to log back in for each visit.
Finally, cookies can be turned off by the user, while session ids can't (unless you delete them from the URL in which case the site will fail anyway.) So we have to choose between preserving state when the user has gone offsite, but can turn that off and screw it up, or prevent the user turning it off and screwing it up, but losing state if they go offsite.
Either way, the site needs to know that it's you who requested a page. That's so we don't go showing your account balance to all and sundry. By disabling cookies, you make our job that much harder. In some cases, you make it impossible.
In essence, what you are doing by disabling cookies on your bank's website is like going into a bank branch without your bank card, passbook or any ID and asking to transact on your account. It isn't going to happen, for obvious reasons.
So please, just enable the cookies, at least for sites you need to log in and preserve state on. It'll save you a lot of headfuck in the long run, and save many a poor web developer from tearing their hair out having to track state with people who insist on refusing to be tracked!