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Chess Program

No, I mean like you can draw arrows and things. The guy was showing all the moves a piece could do. Sorry, I probably said it wrong.
Like with the analysis board if I try a move and I like it then I try a different move and don't like it but I forgot the first string, the board doesn't keep track of the different moves I tried. But with the program that the guy uses in teaching videos he draws arrows showing all the moves he's going to show us about.

I don't know what software made that video... there are several ways to do it (make you tube videos) . I think ChessBase is the most common.
But it sounds to me like you want a program that will let you examine your moves like the analysis board does, but also save all the variations you try. There are zillions of programs that will do this, but at the same time, what you are looking for doesn't exist (or I haven't found it) let me explain:
What doesn't exist is a simple, stand alone .pgn reader/writer that will let you manipulate chess pieces on a virtual chess board annotate moves and variations and save the results.Acutally, this program DOES exist, but only as a java app for a website. So, if you are good with webpages you could build your own little website with chess app and do this. (Unfortunately, I don't know beans about website creation, even simple personal ones.. I should probably change that about myself) ... The app is called Chess Viewer Deluxe.
But there is no simple downloadable small non-system clogging .pgn viewer that I know of... I wish I could just run Chess Viewer Deluxe on my desktop but I don't know how to do it (anyone know how to do this?... or does an equivalent program exist that I can run from my desktop?)
What there are, are big omnibus programs that do lots of things.. and one of the things they do is let you manipulate .pgn files.
Computer chess programs like Fritz and Chessmaster and Arena (a free chess GUI) will let you do it. You download the game's .pgn and open it with Fritz or CM and it will track all your variations. You just have to make sure the analysis engine is turned off or they will give you moves and/or evaluate your moves -- that's cheating. These are also big clunky programs that take a noticeable amount of time to load... which is unfortunate for doing something that requires minimal processing. There's also ChessBase, ChessAssistant, and Chessbase Light and ChessDB -- the first two are big expensive databasing programs, Chessbase Light and ChessDB are free -- these will let you save variations on a .pgn -- I haven't used ChessBase, ChessAssistant, or Chessbase light (will use Chessbase light soon though). I use ChessDB constantly. It doesn't have an analysis engine on, by default ... The only problem with ChessDB is that it highlights the square it thinks you will move a piece to. This is borderline IMO. The highlighted square comes up instantly, and with engines turned off, well, what the hell is the program using to make it's decision? It can't be very strong... At the same time, it _is_ some sort of suggestion and how are you supposed to pretend to yourself you didn't see the highlighting. I don't use chessDB as my virtual chess board for correspondence games because of the highlighting feature... frankly that's probably overly squeamish... but I like to be squeaky clean.
If anyone finds that ideal .pgn program, let me know.
Oh, part two of your question -- In "Online chess" at chess.com (not Live Chess, which is different) Yes you are allowed to use that. You are allowed to use anything that doesn't think for you. You can use any tools that help you think about what to play next. You can use another board, you can use books on openings, you can use the databases to look up how games with your position, or similar positions went... you can use anything so long as it doesn't _think_ //that means no asking other chess players for what they'd do in your shoes, and no using computer engines... and no using endgame tablebases (although, technically, these don't "think", they do tell you exactly how to move and exactly what the result of the move will be.)

The program in the videos is Blitzin. It's a client for connecting to the ICC server, and not really designed for keeping a record of your correspondence games. You are better off using a database program such as Aquarium, Chessbase, the original Chessbase Light, ChessAssistant or Jose, or even GUIs like Shredder Classic, Arena, Fritz or the Chessbase version of Shredder.
For a long time in correspondence games, players have been free to consult books and game archives, take notes, and move the pieces around on the board. Using databases/GUIs to keep track of your moves and analysis is just an extension of that old practice.
On Heisman's topic of how to choose a move, Kotov's "Think Like a Grandmaster" was seminal. Other books like Soltis's "How To Choose A Chess Move," Jon Tisdall's "Improve Your Chess Now" and Nunn's "Secrets of Practical Chess" all have interesting things to say on this important topic.

What doesn't exist is a simple, stand alone .pgn reader/writer that will let you manipulate chess pieces on a virtual chess board annotate moves and variations and save the results.Acutally, this program DOES exist, but only as a java app for a website. So, if you are good with webpages you could build your own little website with chess app and do this. (Unfortunately, I don't know beans about website creation, even simple personal ones.. I should probably change that about myself) ... The app is called Chess Viewer Deluxe.
What about XBoard (UNIX) or its Windows port WinBoard?
I don't use winboard, but I'd guess it works, as Arena does... but winboard isn't built to be a pgn reader editor first and foremost...it's built to be an engine gui primarily (as is arena)... I'm looking for a little program like chess viewer deluxe where it's all about reading/editing a pgn...I don't want the visual/mental clutter of the rest of the engine interface. It's not the end of the world having to use winboard, arena, fritz or chessmaster or anyother engine gui, just not, for me, ideal.

Winboard does not allow you to add variations to the main line, and save them. Arena and Jose are two free programs that can save variations. Arena has a few menu items in German, and Jose is slightly buggy Java, but they work well enough. If you are willing to spend a bit of money, Chesspartner or the Shredder Classic GUI are a bit more user-friendly. You can download, and try Shredder Classic for free for 30 days, and Chesspartner as well I believe.
Incidentally, all of the Chessbase family has those arrows and highlighted squares. I think though that what you really need is a move list that includes variations, so you can look at what happens several moves down the road.
Chess Viewer Deluxe does allow variations, but as far as I can see, you can only see the current main line in the move list. I think it's better to be able to see all the sidelines in one window as you can in Arena, Jose, Chessbase, etc.
I have a two part question:
I've been watching a few videos online to learn more about chess. In the videos they all have a neat software that shows all the possible moves. Like he draws arrows to show.
Here's a video that shows what I am speaking of: http://Video.google.com/videoplay?docid=-1505772075260967251&hl=en
I am wondering what program that is.
The second part of my question is this: Is that allowed to use? I don't have a chess board and if I did, I wonder if that is allowed to work out moves on my own. I'd assume it's allowed because it's not much different than the analysis board but It keeps track of all the moves I am thinking about.
Thanks in forward.
Sophie.