Twitter API can be a major PITA sometimes, but luckily there are modules for Drupal that makes integrating it to your website easy.

Required items:

• Twitter module (this tutorial uses 7.x-5.8 version)
• Oauth module (this tutorial uses 7.x-3.2 version)
• A Twitter account

Steps for installation:

1. Download the above modules
2. Enable Twitter and Oauth modules
3. Go to https://apps.twitter.com/
4. Click 'Create New App'
5. Give Name, Description and Website for this integration
6. Under Callback URL, input 'http://yourdomain.com/twitter/oauth'
7. Click 'Create'
8. In your Drupal site, go to http://yourdomain.com/admin/config/services/twitter/settings
9. In OAuth Consumer key and OAuth Consumer secret, input your app credentials (it's under 'Keys and Access Tokens' tab in Twitter)
10. Click 'Save configuration'
11. Go to Twitter tab in your Drupal site
12. Click 'Go to Twitter to add an authenticated account'
13. Authorize the app with your Twitter credentials
14. Your Drupal site should now have your Twitter profile icon displayed! In that line, tick Tweets checkbox and click 'Save changes'

That's it, your site is now integrated with your Twitter feed! The tweets can be integrated with Views to your site. The Twitter module creates a View automatically that displays the tweets, the default url is http://yourdomain.com/tweets. You can edit this view or create completely new.

You can see the result on this site, the tweets on the right are pulled from my Twitter feed and displayed as a View.

If you don't see any Tweets coming up, remember to run cron in your Drupal site.