The Original Series 1x08, Balance of Terror (remastered), at about 16 minutes and 10 seconds in.
The crew just got a view of the Romulans inside their ship, and were startled to see they look very much like Vulcans. This sparks suspicion in some crew members, given Spock's half-Vulcan heritage.
They're trying to make sense of some Romulan transmissions when this scene takes place. The part of the scene the clip is taken from is in bold-italics.
[The camera is following Kirk as he walks around the bridge. Kirk stops next to the Captain's seat. Uhura is at her station in the background. Lieutenant Stiles is at the helm in the foreground]
Kirk (to Uhura): Decoding?
Uhura: Cryptography is working on it, sir.
Lieutenant Stiles: Give it to Spock.
Kirk walks towards the helmsman.
Kirk: I didn't quite get that, Mr. Stiles.
Kirk stops one step past the helmsman's seat, his back to Stiles.
Lieutenant Stiles: Nothing, sir.
Kirk: Repeat it.
[Camera switches to Spock, at his station.]
Lieutenant Stiles: I was suggesting that Mr. Spock could probably translate it for you, sir.
Kirk: I assume you're complimenting Mr. Spock on his ability to decode.
During Kirk's line, Spock bites his lip on "complimenting" and then looks at Kirk on "decode".
[Camera switches back to Kirk & Stiles, with Uhura still at her station in the background.]
Lieutenant Stiles: I'm not sure, sir.
Kirk walks back to about a step behind the Stiles chair. He then turns the helmsman's chair, so that Stiles looks Kirk in the eye.
Kirk: Well, here's one thing you can be sure of, Mister...
[Camera closes in on just Kirk & Stiles.]
Kirk: ...leave any bigotry in your quarters. There's no room for it on the bridge. Do I make myself clear?
Lieutenant Stiles: You do, sir.