Setting up Windows version (0.9.8)
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PCSX2 is a complex system and in it's not so user-friendly some of its parts. You must have some basic computer knowledge to be able to set it up correctly. If you are an advanced user please skip the following paragraph. As of now there are two versions of an emulator we can focus our attention on: the version with the old GUI and the version with the new GUI. Each version is ultimately the same PCSX2 but in a different envelope. The programs with new GUI are newer and might be more compatible. Before you continue further you must determine your version of PCSX2. The easiest way to determine your version is to run PCSX2 and go to the config-configure or config-general settings menu. If you will see some pretty icons like this or in the upper panel you're definitely using the new GUI version. In the old GUI on the other hand there won't be either icons or upper panel (only the plugin selection).
It's also interesting to know the version of your emulator. Some of your games may work with some versions and may be totally broken with the others. Although the newer the version you'll get the less problems is should contain you must note that some special cases (games) may work a lot better with some older versions. This is because the emulator still is far from being complete and each change in it can lead to the correct emulations for some games yet break the others. You'll need to use PCSX2 forums, search engine or this wiki (if it contains your game) to find the best version needed if your game refuses to run. See some additional information about versioning here.
Note that at the time of writing of this article the new GUI version still lacks some features from the old GUI so it's recommended that you wait until the new GUI version will be aged enough for you to use.
Basic PCSX2 knowledge
You may download PCSX2 for free from many site on the Internet. The archive may contain an installer or just some zipped files. It's pretty much recommended that you start with the installer version (try downloading the newest official from here). This will help you to avoid a lot of troubles with the archivers. However in the case you've downloaded zipped (rared, etc.) version of PCSX2 you'll need to manually unzip the files from the archive (try using this archiver) and place them in any folder of your choice (adding a link to your desktop from the main executable pcsx2.exe will also help you to run your emulator quicker).
Be warned not to erase you emulator folder when you're trying to install the newer version. This folder contains all your ini files (your settings), your quick saves and your memcards. If you will accidentally erase the whole folder you may need to play some of your games anew! Thus to update the emulator with the new version of executable it's better just to download the archive and extract the executable (and maybe plugins) into your main program folder (some executables may be named like pcsx2-r1736.exe so you'll need to rename them into pcsx2.exe or update your desktop link accordingly).
PCSX2 is a plugin oriented program. Your pcsx2.exe is the main processor that can emulate the real PS2 on the PC but it cannot work alone. It needs graphics plugin to show the picture and the sound plugin to play some music or ingame sounds it also needs PAD plugin so you'll be able to play with your keyboard, mouse or gamepad. Moreover there may be more than one plugin of one type so you may choose which is better suited for your game. All plugins are stored (by default) in plugins folder of your main PCSX2 folder. Some plugins like GSDX for example can at times be updated quite often so sometimes you may desire to renew only this plugin. In order to do it you'll need to download the updated version and extract it by using archiver into your plugins folder. The changing of some settings (choosing your new plugin) may also be required after this (see the section about how to set up plugins).
PCSX2 will also require the BIOS from your real PS2 in order for you to play. The PS2 BIOS is a binary file can be obtained from your console. It contains some special routines used by games (like memcard handling support) which are not emulated by PCSX2 internally. The dumping of BIOS is one the hardest stages of your PCSX2 experience. You may get some instructions here and start dumping by downloading the tool from this place or use the official guide here. After your BIOS file is successfully dumped from your PS2 you can copy it into bios folder (by default) of your PCSX2.