OpenID is an open, decentralized standard for authenticating users which can be used for access control, allowing users to log on to different services, e.g. OpenStreetMap, with the same digital identity.
OpenID is an optional, additional way to log-in to the OpenStreetMap website. When using OpenID, you will still need a normal OpenStreetMap Account, but can log-in with your ID rather than needing to enter your username and password.
Using OpenID to log into OSM
In order to use OpenID, you will need to associate your ID with your OSM account. If you already have an account, then you can do it by logging-in and going to your account details page. Beneath the change password fields, you will see the OpenID field where you can enter your ID. If you have trouble adding an OpenID (in particular a Google Account with multiple sign-ins) try this: Enter "https://www.google.com/accounts/o8/id" in the OpenID field of your accounts details page, press "Save Changes", then while at Google's "third party service is requesting permission" open a new tab in the same browser and login to OSM with your username+password, then continue the OpenID login. You can now login to OSM without digging up your password.
For new OSM users you can enter your OpenID on the sign-up form either in addition to a password or instead of one.
Issues with OpenID in OSM
OpenStreetMap is not only a single website, but to some degree consists of a whole conglomerate of different (external) services. E.g. editors, forum or trac. See Accounts for a list of some of these. Many of these services talk to the main OSM via username and password. If you only specify an OpenID, but no password, some of these services may not function anymore. Some services use OAuth though, which allows to communicate without needing a password and should still work with an OpenID only login.
Obtaining an OpenID:
There are many ways of obtaining an OpenID and there is a good chance that you already have one, as e.g google accounts, yahoo! accounts or flickr can act as openid providers. An incomplete list of OpenID providers can be found e.g here or here