You can use the Chatter REST API to access Chatter but you will need a user context with which to access / post information.
http://www.salesforce.com/us/developer/docs/chatterapi/
If your intention is merely to display a read only chatter feed then you can get away with an integration user and use the Chatter REST API to display the feed.
If you however want users to be able to post you will need to take care of the authentication via oauth or similar and then have salesforce users set up for all those who intend accessing.
If the users of this external website and Single Signed into Salesforce you could probably get away with embedding an Iframe which uses their already established session with salesforce to display a Chatter Feed