I often hear the term "deep" thrown around to describe various things, i.e., "deep conversation" or "deep discussions" or "deep friendships." Typically, the term is used in a very positive way, it is a characteristic that we desire things to have. I agree that I want depth in a lot of things, but I have realized that there is a lack of clarity, to me at least, as to what a we mean when we say "deep."
Very often, I think that we usually take "deep" to mean either "serious" or "abstract." When we have a very serious conversation we describe it as deep. When someone throws out an abstract idea in a discussion we call them a deep thinker. But I don't think that is necessarily the best way to look at this. To illustrate, lets first explore what we mean when we describe something as "shallow" as opposed to "deep."
Shallow is usually synonymous with "trite." A trite conversation is one that is has no real importance or significance. Talking about your hair color has no real effect on anyone else beyond some superficial opinion judgment. So that seems a very insignificant thing. I think we can justifiably declare "deep" to be the opposite of "shallow." So it would seem that depth implies that what is being discussed actually has some significance. It has an important impact on someone, it is meaningful. In that sense, depth is not merely serious or abstract. In fact, something serious or abstract may actually be shallow if it is meaningless or insignificant.
So how does this apply to various things? First, we need to realize that depth is probably a relative thing, and by relative, I mean based on the situation. So for instance, a conversation would be deep if the things said had a significant or meaningful impact on their audience. If someone is dealing with a situation, depth would involve giving them good advice as opposed to changing the subject to something that has little impact on their situation. So a deep person would be one who is able to discern what is important in a situation or conversation and able to express things meaningfully and positively affect others. A deep relationship would be one where this occurs mutually, each individual positively affecting the other, whether that be by encouragement, advice, constructive criticism, or some other means.
This being said, depth probably doesn't look the same in every case. I think you can be very light-hearted and funny and have a lot of depth if what you're saying has some sort of meaningful impact. Likewise, I don't think that a "deep conversation" has to be serious in tone or academic in nature provided the things that come out have some significance. The measure of that significance may even be different to different individuals in different circumstances. Basically, depth to me seems to be a combination of discernment (or maybe sensitivity to the situation at hand) and the confidence to apply that discernment to the given situation.