There are many possible meanings of word "deep" one can detect in the common speech. I'll list three good and three bad but I do not pretend the list is anywhere near complete.

1. Very difficult (Fermat-Wiles, Carleson, Szemeredi, etc.). These theorems usually stand as testing tools for our methods and we can measure the development of the field by how easily they can be derived from the "general theory". Their "depth" in this sense deteriorates with time albeit slowly.

2. Ubiquitous (Dirichlet principle, maximum principles of all kinds). They may be easy to prove but form the very basis of all our mathematical thinking. This depth can only grow with time.

3. Influential (Transcedence of $e$, Furstenberg's multiple recurrence). This meaning relates not as much to the statement as to the proof. Some new connection is discerned, some new technical tool becomes available, etc.

1') With an ugly proof (4 color, Kepler's conjecture). They usually reflect our poor understanding of the matter

2') Standard black boxes used without understanding ("By a deep theorem of..." something trivial and requiring no such heavy tool follows). They are used to produced junk papers on a conveyor belt and create high citation records.

3') Hot (I'll abstain from giving an example here to avoid pointless discussions). They reflect the current fashions and self-promotion.

There are many possible meanings of word "deep" one can detect in the common speech. I'll list three good and three bad but I do not pretend the list is anywhere near complete.

1. Very difficult (Fermat-Wiles, Carleson, Szemeredi, etc.) These theorems usually stand as testing tools for our methods and we can measure the development of the field by how easily they can be derived from the "general theory". Their "depth" in this sense deteriorates with time albeit slowly.

2. Ubiquitous (Dirichlet principle, maximum principles of all kinds). They may be easy to prove but form the very basis of all our mathematical thinking. This depth can only grow with time.

3. Influential (Transcedence of $e$, Furstenberg's multiple recurrence.) This meaning relates not as much to the statement as to the proof. Some new connection is discerned, some new technical tool becomes available, etc.

1') With an ugly proof (4 color, Kepler's conjecture). They usually reflect our poor understanding of the matter

2') Standard black boxes used without understanding ("By a deep theorem of..." something trivial and requiring no such heavy tool follows.) They are used to produce junk papers on a conveyor belt and create high citation records.

3') Hot (I'll abstain from giving an example here to avoid pointless discussions). They reflect the current fashions and self-promotion.