If worpdress redirects you back to the login and password page (and not displaying any info that login/password was wrong), this is a sign that the login session is broken.
Most often this is related to cookies. WordPress needs cookies for you to login. If cookies are not properly set or deleted later or are just mis-configured (there are some configuration constants available to control those, see in codex here and here).
You can tackle cookie issues down by using your browser and check which cookies are set or not set. Compare that with the information provided in codex to verify if that is your issue or not.
If cookies are properly created, transfered from server to browser and then back from the browser to the server on the next request, then this is not a cookie issue. Something else then prevents on your blog to keep the login session.
As the wordpress login session relies heavily con cookies, most often such issues are cookie related, because there is not much more to break. But sure, incompatible add-ons can break everything, so as well for that issue.
If you experience it only on one computer with one browser, give the computer a reboot. Probably you just had a hick-up somewhere and this solves stuff. Keep in mind that some firewalls block cookies or referrers. Blocking referrers can create problems to login: http://hakre.wordpress.com/2010/05/26/you-do-not-have-sufficient-permissions-to-access-this-page/ – But normally this creates a different behavior.