How is dialog between two people filmed?

This will be achieved through a series of shots.

Let's pretend you want to show a scene where two people are talking in a diner - here is the classic way to go about it.

First you shoot an establishing shot of the whole room - your actors can perform the whole scene and it doesn't matter if they mess up, as you will not be using the dialogue from this shot any way.

Then you would shoot a mid-shot. This would involve both actors in the frame, shot from the side (or even overhead), and they would perform the whole scene again.

Then you would shoot a series of 'over the shoulder shots'. This is where you place the camera over the shoulder of one character while the other one speaks, then vice versa. One thing to remember is to place the camera over opposite shoulders, otherwise your line of action will be screwed up. You might even notice that actors don't look directly at each other during these shots, that's because an exaggerated eyeline (looking a few inches to the side of the other actor's head) reads better on film.

Next, you might want to shoot a series of 'talking head' shots, where it is just one actor in frame - you would use this for reactions and monologues, then repeat with the other actor.

Finally, all the footage is edited together to create a seamless shot. The audio would generally be taken from the best take - if there is an issue with lip-sync then the editor now has plenty of coverage (extra shots) from which to cut away. Many times you might notice a long shot during a conversation, this is because the audio was perfect but the director didn't get a good shot of the actor. Watch mouths and backs of heads in these shots to see if the lips match the dialogue (not always!).

I've filmed in many different circumstances, and one scene which was a discussion at a dinner table involving six people once had to be shot piece-meal for time constraint reasons. I shot each actor saying their lines (I fed them each line), and then shot a series of establishing and reaction shots so that I could edit the whole thing together. It's not perfect, but that's independent film for you :)

There are lots of resources in YouTube which will teach you how to shoot any scene if you know what to type in the search box. To start google the terms in bold.

If you shoot line by line, when you edit the clips together, you will have an unnaturally choppy conversation.

The common way to do this is by letting the actors play the conversation while shooting just one of them. Then make them repeat the conversation with the exact same words while shooting the other actor. That is called a "Shot-Reverse Shot". After this you will have two clips of video which can be cut perfectly from one to the other and the reactions and silent bits will be smooth and natural.

Very often is useful to make the actors repeat the conversation a third time and having both of them in a wider frame, that is called a "Master Shot".

In YouTube, Tom Antos and Film Riot have good tutorials. If you really want the video to look more professional, learn about audio editing the 180 Degree Rule and the Rule of Thirds at least.

Master shot with both actors showing their relationship to surroundings, each other. Close up of him. Close up of her. Over him to her & over her to him if scene longer than 2 pages, for example. With that footage, and professional actors matching their movements so all the takes from all the angles match the master, then all the closer angles, the overs, will all cut with the master. This is a short, rigid, old fashioned tv version of shooting dialog. Mike Nichols would have differed. So would Billy Friedkin.