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 possible issues with archivers. In case you've downloaded an archived version (ZIPped, RARed, etc.) 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 start your emulator quicker).
Be warned not to erase you emulator's 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 does not work alone. It needs a graphics plugin to display the game graphics and a sound plugin to play game sounds and music, and it also needs a pad plugin to allow you to play using your keyboard, mouse or gamepad. Moreover there may be more than one plugin of each type so you may choose which is better suited for a particular game. All plugins are stored (by default) in the plugins folder of your main PCSX2 folder. Some plugins like GSdx for example can be updated quite often so you may want to renew only this plugin (out of all plugins). In order to do it you'll need to download the updated version and extract it by using the archiver into your plugins folder. The changing of some settings (choosing your new plugin) may also be required (see the section about how to set up plugins).
PCSX2 also needs 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 more complicated stages of your PCSX2 experience. You can refer to the 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 should copy it into the bios folder (by default) of your PCSX2.
The ordinary user may use this to create a shortcut to his/her game on the desktop so that PCSX2 will run the game immediately. To do this make a link to PCSX2 main executable (pcsx2.exe) and update the target field of this link, it will look something like: 2b1af7f3a8