This is indeed a welcome change after Google begun its extended experimentation with displaying Twitter contents in desktop search results. Google started introducing the display of Tweets in search results in Mobile from May, this year and will now show also show it for desktop searches in English language. Google will now make available this service to the global users in English language. The process is similar in both mobile and desktop searches. Just like the mobile searches, they started in May, Google will display Tweets within a carousel placed in the main column showing the organic search results. The Tweets will only be visible if the Google algorithm considers them to be truly relevant. Moreover, users making the search do not need to have a Twitter account mandatory to get the search results or to click on the Twitter results.
Furthermore, Google announced that the rankings and relevance behind these search results from Twitter will be continuously evolving and improved over the time as the process gets matured. At present, there is no provision to show Tweets from advertisement components in the organic search results.
You can personally try and experience the tweets coming in your desktop search results. It will not come for every random search though, and will be seen in keywords with truly having some relevant search results on Twitter.