I have written an application which using twitter authentication.
Authentication process:

Get the request token
Authorize the token
Get the access key

Now I have access key, I can use the twitter API. I stored the access_key pair in the database for further use. But when next time user logged in via twitter, since already i have access_key pair I dont want to authorize. How can I do that?
You mean you're using the "log in via twitter" hack?
Well, if you want a "remember me" option you can build one just like normal (store some session info in a cookie... all security caveats apply).
If the actually are logged out and there's no cookie then you have to go through the whole process again anyway.  The authorize step will just redirect them right back, likely, since they have already authorized your app.
twitter authentication issue using oauth?