quote tweet

A  quote tweet  (also known as a " retweet with comment ") is a reply on Twitter to an original tweet that displays the entirety of the original in a reply-context below the reply text.

Quote tweet is available in the Twitter UI as an option inside their "retweet" button.

Examples
With reply text commentary:
 * https://twitter.com/kfury/status/661738224969121792

reacjweet:
 * https://twitter.com/kevinmarks/status/661802300407545856

Backfeeding
If someone posts a quote tweet in response to a POSSE tweet copy of one of your posts, it makes sense to backfeed it to your original like any other @-reply tweet.

Bridgy support
Bridgy does support backfeeding quoted tweets. Add your examples / opinions to the feature request:
 * https://github.com/snarfed/bridgy/issues/504

Articles
Articles about quote tweets and similar constructs:
 * 2015-05-22 Mentions as quotes