Technically, in correspondence, anything other than the use of engines is allowed. So, even as you are playing the opening, you can look up book moves and opening theory, in the middle game you can find and analyse positions in other games, and in the endgame you can refer to end game theory (though not tablebases). Just don't use engines to calculate best moves for your position.
If general studying were not allowed, then I would never study, because I always have online games on the go now :p
Clarification
These days, they usually do, yes. That's ok - just don't fire up Houdini and plug your position in, and you'll be fine.
Consider the question from the flip side: you sign up for a 14-day per move tournament. Half your opponents go on vacation once they start losing, and the tourney takes a year to complete. (This is not far off the mark for this site, actually). For a year you couldn't pick up a chess book or do a tactics puzzle in the newpaper??? You would deprive yourself of quite a lot, needlessly, by subscribing to the puritanical ethos you are asking about.
Generally speaking, if something else is telling you the indisputably best move for your specific position, then it isn't allowed. (Note that I am only speaking of Online Chess here, not Live Chess. In Live, there are further restrictions.)
Opening books/articles and databases are ok because their suggestions are generally not indisputably best. After 1.e4 c5 2.Nf3, there are many playable moves. While a certain move may be the most popular (has many played games and a good win%), a different move may be better for you and still be perfectly playable even at the highest levels.
Endgame books/articles are allowed because they are a static resource that doesn't give you the best move for your specific position (unless you have achieved theoretical positions such as the Lucena position), but rather give you an idea of the technique you need. The specific moves are still up to you.
Tablebases and engines are not allowed because the suggestions they give you are the best move for your position.
The analysis board is ok because it is just a visual aid (much like a real life board you might set up at home); all of the analysis is still up to you to perform.
Things like tactics trainer most likely won't give you the same position as in your games, so using it while you have games going is ok. Even if you coincidentally get a similar position, the differences are likely to be enough that you still need to do fresh calculations on your own for your game.
And as ThePeanutMonster said, if studying weren't allowed, then a large number of members on this site wouldn't be able to study at all, and having tools to study is one of the big draws of this site for many people (myself included). Anyone (premium or free) can join as many of the official tournaments as they want, and these start every month. Since most games will take longer than a month to play, it is entirely possible to be on this site for months or even years at a time and have at least one game going at all times. I can't recall the last time that I had zero games going.
How does anybody know that a person is using tablebases? What are the restrictions of resource for live?
Are we allowed to do chess training while we have correspondence games going on ? For instance, I have some correspondence turn based games going on (you know NOT on the live chess portion of the site) and I did tactics trainer the other day (possibly yesterday) then I felt a little guilty because I did not know if this was technically cheating. I mean I don't think it is, I believe on this site people read articles, practice tactics, etc. (not during live games of course), so I was wondering if I am doing everything okay and within the rules? I can see how it would help me during my current games but in correspondes you can use the analyzer and analyze the game yourself, and I'm sure people get some help from chess articles, keeping up with current games, etc. At the time I didn't think I was doing anything wrong, but I just want to make sure I am allowed to study chess while I have correspondence games (just the regular games) going on? (I am not like looking for my positions in other games or anyting, just practicing chess and trying to get better, I realize that this is not okay in regular face to face chess and live chess, but I'm thinking correspondence chess is different).