Good topic ... everyone learns that they should dive with a buddy, and so it behooves us to understand what that means ... both for ourselves and the person(s) we choose to dive with.
Let's start with a fundamental concept ... diving with someone else means that we need to be able to communicate. In scuba diving, this has several aspects ... many of which are unfamiliar to us at first, because we can't talk to each other underwater.
Communication starts with the dive plan. Both of you should be informed of what it is ... if not directly involved in creating the plan. Both of you should know ... before you enter the water ... where you're going, how deep, for how long, and what the expectations of the dive are. Having a good dive plan, and making sure that everyone who's involved in the dive understands what it is, sets the expectations for what the dive will involve. It puts everyone on the same page before you get in the water ... while there's still a chance to talk about it and answer questions.
Now, things can change once the dive gets underwater ... which is why communication underwater is also important. So once the dive gets underway you need to be able to see your dive buddy, and vice versa. This puts positioning ... swimming to be seen ... pretty much at the top of the buddy skills list. On most open water dives, swimming to be seen means side-by-side ... where all it takes to see each other is a turn of the head.
The most basic form of communication underwater is hand signals ... and developing a good lexicon of hand signals helps divers interact better. If you're diving with an unfamiliar buddy, going over hand signals together should be part of the pre-dive briefing ... so that when you use or encounter signals underwater you know what they mean. There are other means of underwater communication as well ... dive slates or wetnotes to write on, and in some cases using dive lights to signal each other. The latter is particularly useful in night or low-visibility conditions to keep track of each other. Dive lights have a limited range, so if you occasionally place your light in a manner that your dive buddy can see it without effort, it tells him that you're there. We call this "passive communication", and for those of us who frequently dive in low-vis conditions it's a very effective way to keep track of each other.
Diving with a buddy takes a certain mentality ... you have to think of the dive as "our" dive, rather than "my" dive ... because sometimes you'll have to modify what you want to do in order to accommodate your dive buddy. This is a matter of priorities ... and sometimes it can lead to interesting "conversations" underwater ... but in order to be a good buddy you need to put the team ahead of yourself. An example of this would be if you saw something off to the side you wanted to explore, before heading in that direction you need to signal your buddy, get their attention, and point out where you want to go. Either both of you need to go together or you don't go. This is putting team first, and self second. It's a mindset ... any time you want to deviate from the dive plan ... or even just stop and look at something ... you need to think of your buddy first, and make sure you're doing it together.
I have an article on my website that might explain things a bit more ...
NWGratefulDiver.com
... Bob (Grateful Diver)