Go to Extensions.joomla.org and enter "twitter" in the search field. Press "Enter" to see a list of Twitter extensions for Joomla; as of the date of publication, 380 appear. Browse through them and read their descriptions to find one that includes the options and functionality you want for your commenting system. For example, if you want commentators to log in to your site using their Twitter accounts, look for an extension that has a Twitter account-linking function. If you just want users to add their Tweets as comments, choose the extension with that functionality.
If you want to incorporate tweets into the commenting system on your Joomla articles, you need to install a Joomla extension that can access Twitter. Once you have the extension installed, you can determine how users will interact with Twitter on your site. For example, you might only want to treat Tweets about your content as pingbacks in the comments. You have many different options depending on which extension you install.
Download the extension you want.
Log in to your Joomla administrative control panel. Click "Extensions" and "Install/Uninstall."
Click "Browse" and locate the extension's zip file on your hard drive.
Click "Upload File" and "Install." Configure the extension for the options that you want according to its directions.