Illustrations for this article were graciously supplied by David Weber.

An excellent resource to reference on this subject is William Nealy's book, linked below.  The section on hydrotopology is excellent, well-illustrated, and more comprehensive than this article.  In addition, the whole book is an engaging and educational read- I highly recommend it.

I have borrowed much of Nealy's terminology in composing this article, largely because it both makes sense and has become accepted jargon among boaters.

Kayak : The Animated Manual of Intermediate and Advanced Whitewater Technique by William Nealy

This book should be available in most paddle shops.

Things to note:

Water moves fastest in the trough.  The faster the water, the deeper the trough.

Water slows down on it's way up the face, and conversely accelerates on it's way into the trough.  The steeper the face, the more pronounced the acceleration and deceleration will be.

Because water changes direction in the wave (moving up the face, and down the back) bracing and rolling in the wave will require an awareness of this- if you're bracing deep into the face of the wave, be aware that as you reach through to the back side of it, you'll experience a 'downdraft', which you'll catch up with as you enter the next trough.

A partial-depth hydraulic is another expression of the same thing, but with a major difference- in this case, the flow out is slower than the flow in, causing the wave to build up too high- which then causes it to collapse.  It is, in essence, a wave that can't sustain it's own weight.  Because water seeks its own level, the wave will build up more and more until it collapses on itself, spilling both upstream and downstream. As a result, somewhere near half of the top part of the wave will fall down it's own face. As this pile becomes more and more substantial, we begin to regard it as a 'partial depth hydraulic'- essentially, it's a mass of water falling upstream, on top of a mass of water moving downstream.  These sorts of hydraulics can vary greatly in terms of their 'friendliness', in the sense that bigger ones can be very powerful and retentive, and also in the sense that your comfort level is an individual thing.  In general, however, these types of hydraulics tend to be flushy, provided you can manage to get under the pile.  Swimmers generally flush immediately, but objects that float (such as you in your boat) may or may not, depending upon the hole.  Partial-depth hydraulics tend to make the best playspots, and because they will flush a swimmer, are not generally regarded as dangerous in and of themselves.

Things to note

Outflow is generally not very aerated; rather, look to see a violent boil making up the backside of the wave.  Backflow tends to be of similar nature- large blobs of water cascading down the face tend to form the pocket.

The behavior of this kind of hydraulic will vary with respect to the depth and relative speed of the pool behind the wave, the speed of the inflow, whether there are downstream obstacles backing up the feature, and a host of other details- these can vary in temperament from gentle and easy to violent and insane.

You'll note that on the backside of these features that, like the wave, the collapsing downdraft on the backside can provide unpredictable purchase for braces and rolls.

A full-depth hydraulic uses the same principles, but behaves slightly differently. If the initial downflow is powerful enough to punch straight to the bottom, a different phenomenon occurs- a full-depth hydraulic. Essentially, this is what you'd get if you sprayed a fire-hose into a barrel- the stream would go straight to the bottom and scatter from there, and the water in the pool would then get sucked back into the downflow, causing a very violent experience.  Full-depth hydraulics tend to be unstable and surgy, pushing things caught in them out, or else (in rare cases where there is a sieve) pinning them to the bottom.  Because the current is crashing into the bottom of the river, the outflow tends to be forceful, shallow, and directed downstream.

Things to note

Outflow is generally highly aerated and explosive- landing in these features from above can result in going relatively deep.

These things are loud and scary-looking.

Rolling or bracing in these features can be difficult due to the highly-aerated nature of the water- don't expect solid paddle purchase unless you're engaging the outflow.

A vertical pourover tends to vary in danger in proportion to the depths of its pool, and the shape of its outflow. If it's shallow, or if not a lot of water is pushing back upstream, these hydraulics tend to be very violent but flushy- in general, they are full-depth hydraulics. If there's a flat backflow in a deep pool, these can be highly stable, uniform hydraulics that are exceedingly difficult to escape.  These are typified by a stable crease, a steep entry, a distant boil line, and often a crowning outflow, rather than an outflow directed downstream.  Vertical pourovers are not recommended places to play, both because escapability can be a factor and maneuvering room in them can be limited.

Things to note-

Pool and backflow tend to be highly aerated

Backflow that is flat indicates a large mass of water backing up into the feature- these are very dangerous

On this type of hydraulic it is vitally important not only to look for the direction of the outflow, but also to note the gradient of it as well.  In bigger water or in tight quarters, vertical pourovers can super-elevate the downstream water, making it very tough to escape- you'll need to paddle uphill to avoid being fed into the pourover, and this can occur even if the outflow is going downstream.  The author had to learn this the hard way, you do not.

Rolling in this type of feature can be exceptionally difficult, due to the fact that in the crease, both directions constitute essentially a roll against oncoming current.

Smiling vs. Frowning: In general, if the corners of the hole are pointing upstream, you'll have to go upstream (not an easy trick) in order to get out- that, or you'll have to sub under the pile at least as far as the boil line, lest ye face another round in the hole. If the corners of the hole are pointing downstream, the hole will tend to 'shrug' you off of it's shoulders.  We sometimes refer to these as 'smiling' or 'frowning' features, based on the way they look from upstream- if from your upstream perspective the ends of a hole are 'frowning', it means that the shoulders of the hole are upstream and the hole may be difficult to escape.

Directional Kick:  Often, when a hydraulic is not perfectly perpendicular to the flow of the river, the pile will 'roll' to one side or another, and will have the effect of 'kicking' anyone or anything that hits it to one side or another.  Recognizing this kick is important not just because it gives you a better idea of how to hit it without capsizing, it also tells you that the wave is diagonal to the incoming current and you'll likely only be able to surf in that direction.  This is especially important to note when you spot from upstream a pile whose sides both kick to an unbroken center, because it indicates a hydraulic you probably won't be able to exit on either side... you'll have to exit by either going over the top or underneath it.  (Incidentally, the most effective way to punch these waves is by hitting them in such a way that your boat and inertia are square to the direction of the kick.  For very diagonal waves with a sharp center, often the best place to punch is in the center because the likelihood of being redirected by the oncoming water is lower.  In situations where the diagonals don't form a sharp angle, however, often the hydraulic is at its worst here and punching through a shoulder is a better bet.)

One thing that'll help you identify the difference between a 'wave' and a 'hole' is where the boil line is. With practice, it'll become the first thing you notice when looking at a water feature. The boil line is where the water moving upstream separates from the water moving downstream- it's that spot where the upwelling caused by the fast-moving downflow actually comes up, and has to decide whether to collapse upstream or downstream.  On a wave, there is no boil line- all of the water that goes up to the crest of the wave goes down the back side.  On a hole, some of that water collapses down the front of the wave. 

Most of us have looked at that innocent-looking lowhead dam, thinking, "I've run tougher drops than that.. why am I portaging?" ...and then noticed that the boil line is 30 feet downstream of the crease. In general, the farther away the boil line, the greater the mass of water is moving upstream into the crease, and the stickier the hole will be.

The significance of this is that if you're on top of the water, the river will be moving you into the crease, a subduction zone where both upstream and downstream surface water will push you and the only way out is down. If you're a floaty object and by that nature 'down' is not really a sustainable option, chances are you'll stick there. This is good if you're playing and want to be there, bad if you're having troubles and want to escape. In this respect, the amount of 'float' your boat has (this is usually expressed in terms of displacement volume) will often determine how 'retentive' the hydraulic will be for you.  In general, more volume means more retentiveness- meaning that for escaping big holes, big boats are not necessarily a good thing.

Things to note when you look at a hydraulic: