Depth is what you do not see immediately at the surface of things. Depth is what is below that surface: a body of water below the surface of a lake, the rich life of a soil below the dirt or the spectacular line of reasoning behind a simple statement.
Depth is a straightforward aspect of the physical world. Gravity stacks stuff and not everything can be at the top. Below there is more and you can dig for it.
Depth acquired a particular meaning with the rise of complexity science a quarter of a century ago: What is characteristic of something complex? Very orderly things like crystals are not complex. They are simple. Very messy things like a pile of litter are very difficult to describe: They hold a lot of information. Information is a measure of how difficult something is to describe. Disorder has a high information content and order has a low one. All the interesting stuff in life is in-between: Living creatures, thoughts and conversations. Not a lot of information, but neither a little. So information content does not lead us to what is interesting or complex. The marker is rather the information that is not there, but was somehow involved in creating the object of interest. The history of the object is more relevant than the object itself, if we want to pin-point what is interesting to us.
It is not the informational surface of the thing, but its informational depth that attracts our curiosity. It took a lot to bring it here, before our eyes. It is not what is there, but what used to be there, that matters. Depth is about that.
The concept of depth in complexity science was expressed in different ways: You could talk about the actual amount of physical information that was involved in bringing about something — the thermodynamic depth — or the amount of computation it took to arrive at a result— the logical depth. Both express the notion that the process behind is more important than the eventual product.
This idea can also be applied to human communication.
When you say "yes" at a wedding it (hopefully) re-presents a huge amount of conversation, coexistence and fun that you have had with that other person present. And a lot of reflection upon it. There is not a lot of information in the "yes" (one bit, actually), but the statement has depth. Most conversational statements have some kind of depth: There is more than meets the ear, something that happened between the ears of the person talking — before a statement was made. When you understand the statement, the meaning of what is being said, you "dig it", you get the depth, what is below and behind. What is not said, but meant — the exformation content, information processed and thrown away before the actual production of explicit information.
2 + 2 = 4. This is a simple computation. The result, 4, hold less information than the problem, 2 + 2 (essentially because the problem could also have been 3 + 1 and yet the result would still be 4). Computation is wonderful as a method for throwing away information, getting rid of it. You do computations to ignore all the details, to get an overview, an abstraction, a result.
What you want is a way to distinguish between a very deep "yes" and a very shallow one: Did the guy actually think about what he said? Was the result 4 actually the result of a meaningful calculation? Is there in fact water below that surface? Does it have depth?
Most human interaction is about that question: Is this bluff or for real? Is there sincere depth in the affection? Does the result stem from intense analysis or is it just an estimate? Is there anything between the lines?
Signaling is all about this question: fake or depth? In biology the past few decades have seen the rise of studies of how animals prove to each other that there is depth behind the signal. The handicap principle of sexual selection is about a way to prove that you signal has depth: If a peacock has long, spectacular feathers it proves that it can survive its predators even though the fancy plumage represents a disadvantage, a handicap. Hence, the peahen can know that the individual displaying the huge tail is a strong one, or else it could not survive with that extreme tail.
Amongst humans you have what economists call costly signals: Ways to show that you have something of value. The phenomenon conspicuous consumption was observed by sociologist Thorstein Veblen already in 1899: If you want to prove that you have a lot of money, you have to waste them. That is: Use them in a way that is absurd and idiotic, because only the rich guy can do so. But do it conspicuously, so that other people will know. Waste is a costly signal of the depth of pile of money. Poor people have to use their money in functional way.
Handicaps, costly signals, intense eye contact and rhetorical gestures are all about proving that what seems so simple really has a lot of depth.
That is also the point with abstractions: We want them to be shorthand for a lot of information that was digested in the process leading to the use of the abstraction, but is not present when we use it. Such abstractions have depth. We love them. Other abstraction have no depth. They are shallow and just used to impress the other guy. They do not help us. We hate them.
Intellectual life is very much about the ability to distinguish between the shallow and the deep abstractions. You need to know if there is any depth before you make that headlong dive and jump into it.