Twitter has recently introduced an option of embedding any tweet in our site.
When you hover over a particular tweet. It will show an option called more.
From more you will get two options
a) Email Tweet
b) Embed Tweet
When you select embed, there will be a pop up with the link.
Copy the link to your site and include what ever tweet you like with your website.
I liked the following tweet and I have embed it in this site with the following link.
— Aircel (@Aircel) May 21, 2013