(A lot of what follows is taken from Wikipedia.)
Snow is just frozen water with air in it. Some snow has more air, some less, until you get down to zero and then you've got ... ice.
So let's start with
Powder is just freshly fallen snow -- that's it! As Dan Egan said, I think, from a skiing point of view powder is anything that comes up over the edges of your skis. But, of course, there's powder and powder. Some powder is fresh, some not so fresh. Some is shallow, some is deep. When powder is so deep that skis don't sink to something harder underneath you can call it bottomless powder.
Then there is the density of powder. Here is Wikipedia:
"The water equivalent of the snow is the thickness of a layer of water having the same content. For example, if the snow covering a given area has a water equivalent of 50 centimetres (20 in), then it will melt into a pool of water 50 centimetres (20 in) deep covering the same area. This is a much more useful measurement to hydrologists than snow depth, as the density of cool freshly fallen snow widely varies. ...
... The density and moisture content of powder snow can vary widely; snowfall in coastal regions and areas with higher humidity is usually heavier than a similar depth of snowfall in an arid or continental region. Light, dry (low moisture content, typically 4 - 7% water content) powder snow is prized by skiers and snowboarders. It is often found in the Rocky Mountains of North America and in Niseko, Japan."
After powder has been around for a while, it gets to be tracked or chopped powder -- by being skied over.
When snow has been skied over a lot -- or rolled over by a piste machine -- it can become
packed powder, hard-pack or groomer
which is the snow you'll mostly ski when starting skiing. Variations are
which I define as the rail-like horizontal ruts left by grooming machines, and
which, "when used in the East Coast sense of the word, typically means ice not just firm packed powder. As snow falls, melts, freezes, is groomed repeatedly, is skied on and scraped off from the hardest layer, melts, and freezes again it creates glare ice. When people around here describe conditions as boilerplate this is typically what they mean. This type of 'snow' is generally not found west of the Mississippi. At least not in the quantities and frequencies that it is on the right coast." (JayPowHound).
are formed as the natural consequence of skiers following each other's paths down the slope. They're bumps in the snow that you have to learn to ski round (or over!) Nowadays, resorts groom them away on most of their trails.
Once you get outside the trails there are, as you would expect, more variations.
is a layer of hard snow (often caused by melting and re-freezing or by the compacting effect of wind) over soft snow. It can be a nightmare to ski.
(or spring snow, as it's called in Europe) is delightful, on the other hand. When the sun gets on a firmly frozen slope and melts a top layer of crystals the result is a creamy surface that's wonderfully skiable. But you need to time it right!
Here's what Wikipedia says:
"This covers varieties of snow that all but advanced skiers find impassable. Subtypes are (a) windblown powder with irregularly shaped crust patches and ridges, (b) heavy tracked spring snow re-frozen to leave a deeply rutted surface strewn with loose blocks, (c) a deep layer of heavy snow saturated by rain (although this may go by another term)."
You get the picture, I guess -- pretty much unskiable stuff. But I often read posts on EpicSki reviewing skis which talk about "crud-busting" and "powering through crud". I assume that what people are talking about is loose powder lying on the trail -- but, to be honest, I don't know.
Please, anyone who knows better about this or anything else, correct me!