I've been drafting heavily for over two years now, and thought a lot about various formats and some other attributes that define them. These categorizations might be able to help you adjust to a particular format, but in some cases they are simply qualitative measures that may make you favor a particular format over another.
This designation is somewhat similar to prince/pauper, since it deals with the distribution of power level of cards in the set, but the depth of a particular format has to do less with the difference in power level between the bombs and the commons and more to do with how quickly the power level declines among the commons.
A deep format is one in which card quality remains relatively high even towards the end of packs, meaning there are probably still playable cards among the last three picks. A shallow format, however, sees a dramatic falloff in card quality soon after the best cards are taken, so that typically in the last half of a pack, finding desirable cards is very difficult.
An example of a deep format is Innistrad. In the same block, Avacyn Restored was a shallow format. I would often end an Innistrad draft with 27 or more playables, and the difficult decision was which cards to cut. AVR was the opposite, and I'd be scrambling for playables by the end of the draft. During deckbuilding I'd have to decide which cards were the best of the worst to include. Both situations require useful skills. They both involve maximizing the value of your pool, but shallow formats are more unforgiving, especially in the latter part of drafting if you're not mindful about what your deck needs and you panic and make bad decisions like moving into another color to strain your mana base.
Of course, some of these designations aren't mutually exclusive. When we talk about the speed of the format, we could be washing out information by just talking about the average speed. A given format may support decks at both ends of the spectrum, in which case talking about the average speed (e.g. on average most games end on turn 7.2) is not very useful.
The current format as of this writing, Theros, is a good example. It's not all that useful to talk about whether the format is fast or slow. What is useful to know is that the format supports some very fast decks, and so your grindy, powerful deck needs to be equipped to deal with them or you could be losing a lot of matchup-dependent games. For example, in one of my first Theros drafts, I had a very powerful blue-black deck, with four Grey Merchants. I knew the power of the card, even early in the format, and I expected to be able to come from behind against even the most aggressive starts. In round one, I faced a red heroic deck that turn 1'd an Akroan Crusader, then used two heroic triggers on turn 2, and was swinging for 10 on turn 3. I had no defenses in place and got completely run over.
In subsequent Theros drafts, when drafting a slower, more controlling deck, I began to prioritize early blockers in my colors, especially the two cheap deathtouch creatures, Baleful Eidolon and Sedge Scorpion. Cheap defenders like Returned Phalanx were also very important to fend off the super-aggressive decks and buying time to play very powerful cards like Grey Merchant.
In the last format, M14, the lower overall power of creatures meant that you were unlikely to be facing enormous pressure early, and people started to figure out that the expensive card-advantage spells like Opportunity were much better because of it. The most aggressive deck was probably slivers, but that was a very difficult archetype to draft, since the slivers were good on their own and it was difficult to draft a critical mass of them. So unlike Theros, making sure you had answers to very early aggression wasn't quite as important.
The MtG design team usually does a decent job of balancing the card quality among the colors, but it's a difficult thing to get just right and sometimes the format just isn't balanced. Avacyn Restored is a good example, with blue, green, and red being far better than white or black. Black was also in the unfortunate position of getting a rather weak mechanic (the so-called "loner" mechanic), in lieu of any soulbond, which was a very strong mechanic. Even though black had the best removal on average, and a few standout cards, it was usually avoided. This led to a sub-strategy where a careful drafter might be able to exploit black's undesirability and soak up all the black cards, making a reasonably powerful deck. Some players swore by this strategy, but I rarely saw it in action in practice. A lot of the better black cards, like Homicidal Seclusion and Killing Glare, were easily splashable, so strong drafts often cut them, making the lone black deck at the table significantly worse.
If there is a quality disparity, and you can identify it early, you might have a significant advantage over fellow drafters. As a format wears on, though, that advantage is going to diminish. There might be cycling strategies where, as in Avacyn, you can prioritize the weaker color in an effort to get a strong pool with the best cards in the worst color, but again, I have usually not found this to be a good strategy.
Some formats lead you down certain paths, and punish you from deviating from them. I call these formats "constrained". The more constrained, the fewer archetypes the format supports. The most obvious example are the recent Return to Ravnica formats. The first two sets basically forced the drafter into one of the five guilds. If you were drafting RTR and decided to draft a non-guild color combination (such as UG), your deck was probably going to be worse than most guild decks, which were rewarded with powerful aligned gold cards and high synergy between the aligned colors. Same with the following set, Gatecrash.
Another constrained recent set was Modern Masters, which sported very rigidly-defined archetypes (Giants, Faeries, Affinity, Rebels, etc.). Deviating from one of the main archetypes usually led to sub-par pools, since the cards in those archetypes were highly synergistic.
A good example of a free format was M13, in which the colors were very evenly balanced and a strong deck could be built out of nearly every two-color combination. The result is a format that generally rewards repeat play. I found myself getting bored more quickly with the RTR drafts because there were basically five decks you could build, and you usually either got a very strong representative of that archetype, or you were getting cut and got a very bad deck in your guild.
One thing I didn't discuss was the average power level of the format (e.g. Cubes often have extremely high average power levels). Classifications based on relatively power level (e.g. between the rares and commons as in prince/pauper, among the commons as in deep/shallow, or between colors as in balanced/unbalanced) is more useful.
I personally prefer deeper, balanced, freer formats, though I did enjoy drafting the RTR formats and Modern Masters. Avacyn Restored was probably my least favorite format over the last two years, since it was a very princely format (cards like Bonfire and Entreat were basically unbeatable), very shallow, and unbalanced.
The current format, Theros, is quite good, though. The average card quality is high, and it supports many different archetypes, of varying speeds. Grey Merchant should totally be an uncommon, though.