Here's a zillion year old article that I wrote that may help out. It only covers SD because HD wasn't out yet.
A bunch of years ago the folks that figured out how to broadcast moving pictures came up with a formula for breaking up the stream of voltages that create a TV picture into lines traced on a illuminating phosphor. It involved time and distance. When we started making digital images out of the analogue information the rows of pixels we needed to create was established by the number of scan lines in a TV signal. Even though there were some arguments the standard number of rows for NTSC was 486 (because that's the number of scan lines in a tv signal that, most of the time, contain valid picture information, and 526 for PAL signals. Now came the problem of dealing with the horizontal division of the signal into columns. It was determined because of the math involved that a number that would work for both NTSC and PAL was 720. Thus a standard television picture was determined to have 486 rows and 720 columns of pixels. The only problem was that the aspect ratio of a TV image is 4:3 and the aspect ratio of this block of pixels was not. Rectangular or non square pixels were invented.
In reality, there's no such thing as a square pixel or a non square pixel. A pixel is just a bunch of numbers describing a color value. The reason we didn't go with the right number of columns for a proper 4:3 aspect ratio was that the human eye makes up the softness in the horizontal, the data rate was less, and it worked OK.
Fast forward to HD. When it was first proposed the systems used to process HD images were thought to be sufficiently powerful to avoid this problem and we would all be working with square pixels from now on. Oops, Cameras couldn't keep up so several different pixel aspect ratios and reduced frame sizes were contrived to reduce the data rate to an acceptable level.
Enter After Effects and it's handling of pixel aspect ratios. Throw in the requirement that all video displayed on a computer monitor or mobile device should be (not necessarily must be) square pixels because it's more efficient and you've got potential confusion for the users.
The rules are really very simple, and a good and efficient workflow is easy to work out. Here's the first rule. AE automatically looks at the frame size (x pixels wide by x pixels high) and tries to determine what the proper pixel aspect ratio of the image should be. It follows the rules and assigns the PAR that it assumes is correct. In the case of video footage, this assumption is correct every time if the footage came from a camera. If the footage was rendered or created in another application, AE could be wrong. If the image is a photograph or illustration and the frame size is a standard non square pixel frame size it's wrong most of the time for the photograph and all of the time for a vector illustration.
To keep thing from distorting you MUST know what the pixel aspect ratio of your source material is and make sure that it is interpreted correctly. You don't mess with the PAR interpretation for video unless you know for a fact that it is wrong and it didn't come from a camera.
You do question every vector illustration you get and look hard at every photograph or Photoshop file.
It's almost, and I mean 99% of the time, more efficient to do all of your After Effects work in Square Pixel compositions. The software will automatically correct all footage of any pixel aspect ratio and remove the distortion. The only exception to this rule is when you must do motion tracking with any of the built in tools in AE. The Frame size of the Comp containing the footage to be tracked must be the same frame size as the footage. IOW, if you have some HDV footage (1440 X 1080 PAR 1.33) and you're going to Camera Track or Warp Stabilize the footage the comp must be the same size and PAR and frame rate.
If you're not going to motion track the footage then putting HDV footage in a HD comp (1920 X 1080) works just fine. The footage is scaled in the background to fill the HD frame and everything lines up for efficient processing, you're seeing the actual pixels AE will generate, and there is no distortion.
So much for PAR, now on to Aspect Ratio. That's just the ratio between the height and width of an image. It can be anything you like but if you are rendering to any of the dozens of MPEG based codecs, or any specifically designed for video codecs there are very specific rules for frame size that must be followed. H.264 will just stretch your comp to fit the nearest legal frame size. This distorts and softens the image. Others, like trying to render NSTC DV with the wrong frame size, just results in a failure.
I hope this helps. The explanation is actually part of a large collection of articles I'm preparing for publication.