Liveries allow you to virtually retexture FSX aeroplanes with paint jobs --- representing different models or airlines without having to build a completely new 3-dimensional model. That means if you're looking to fly a jet in the colours of your favourite airline, you only need to download a few megabytes of new texture files. Installing liveries in FSX is a matter of moving those files in the proper directory and updating the aircraft's configuration file, a plain text document you can edit without any special tools.
Extract the archive containing the livery; you should see a folder beginning with "texture" and ending with the name of the new livery in the form "texture.name." For example, a livery for the fictional Oceanic Airlines might have a folder called "texture.oceanic."
Open Windows Explorer, and browse to your Flight Simulator X directory --- by default, if you haven't selected a new one, Microsoft installs FSX to "C:\Program Files\Microsoft Games\Flight Simulator X." Inside this directory, browse to "SimObjects," then "Airplanes," and then the aeroplane the livery is for. In general, the folders will reflect the name of the aeroplane with all the spaces converted to underscores; for instance, the Airbus A321 is located in the "Airbus_A321" directory.
Move the "texture.name" directory from where you extracted the archive to the aeroplane's folder by dragging the entire directory into the Explorer window viewing the aeroplane's files. You should now see a series of folders: "Model," "panel," "sound," followed by "Texture," "texture.2," "texture.3," and so on, ending with the new texture folder.
Right-click on "aircraft.cfg," located in the aeroplane's root directory and choose "Open With." Select "WordPad" from the list of recommended programs and then click "OK."
Scroll through the sections beginning with "[fltsim.n]," where "n" is a number starting at 0 and increasing by 1 each time ([fltsim.0], [fltsim.1], [fltsim.2] and so on). Keep scrolling until you find the start of the "[General]" section; when you have done so, select the text of the last [fltsim.n] block and copy it by hitting "Control" and "C" simultaneously.
Hit "Control" and "V" to paste the text; you should have two identical copies of an 18-line block of text beginning with "[fltsim.n]," where "n" is a number and ending with "description=" followed by a boilerplate description of the airplane. Change "n" to one higher than the current number; if the last entry had been [fltsim.5], your new livery will be [fltsim.6].
Change the text following "ui_variation=" to something unique and descriptive -- this will be what you click on to choose the aircraft's livery in FSX. For the fictional Oceanic airliner, you might change this line to:
Replace the text following "texture=" with the ending name of the texture folder; if the livery came in a folder called "texture.oceanic," you would write "texture=oceanic" in the aircraft.cfg file. This tells FSX what folder to look for the new livery's textures in.
Save the file and exit WordPad. When you start Flight Simulator X, you will be able to choose your new livery from the "Select Aircraft" window when the "Show all variations" box is checked.