The View → Pixel Aspect Ratio setting in Photoshop simulates non-square (elongated, rectangular) pixels on a square-pixel screen, primarily for preview purposes.
Photoshop does this simply by scaling the work area along one of the axes to get the desired, simulated pixel shape. The scaling takes place for display purposes only; when you change the pixel aspect ratio, the software will not touch the underlying pixel data in the image you’re working on.
The image resolution (number of pixels along the horizontal axis and number of pixels along the vertical axis) will stay the same regardless of whether you’re watching it in an aspect ratio-corrected mode or in a square-pixel mode. If you set a non-1:1 pixel aspect ratio and use the magnifier tool to zoom into a level which will show you the individual pixels as a grid, you will see that the cells of this grid are now elongated along one of the axes, following the x/y pixel aspect ratio you set.
However, Photoshop does allow you to paint on the image in this mode, and will scale the output of its tools accordingly, to match the new pixel aspect ratio. So you can e.g. draw circles which will look perfect with no distortion whatsoever, even though when you study them in the magnifier view, or using the ruler tool (set to use pixel units), there will be a different number of pixels along the horizontal and vertical axes.
So why would you ever want to do this? Your pixels are supposed to be neat and square; their width matching their height, right?
As the preset options in the View → Pixel Aspect Ratio menu suggest, Photoshop mainly implements this feature for working with video frames. There are several industry-standard digital video formats – such as those used on PAL and NTSC DVDs and in SD resolution digital TV broadcasts – which for technical and historical reasons employ a different pixel aspect ratio than 1:1.
The same also holds true for the early (1980s-era) home and office computers and video game consoles. The early video graphics chips usually produced signal where the pixels – realized as a video raster displayed on a CRT screen – were clearly wider or narrower than their height. If you wanted your computer to draw perfect circles instead of elongated ellipsoids, or design any other sort of graphics or art which was to be displayed on the computer screen, you needed to take the pixel aspect ratio in account and match your designs to the fundamental characteristics of the video graphics modes your computer could produce.
Later on, PCs began to standardize on graphics modes which would produce (nominally) 1:1-shaped pixels on properly adjusted CRT screens, while also filling the screen area from edge to edge. Yet later, LCD monitors fixed the pixel array once and for all, making it (for all practical purposes) mandatory to use square-pixel graphics modes and the native resolution of the display, instead of some arbitrary resolution.
This was all sensible and welcome development as standardizing on square pixels made it much easier to create and display graphics in a portable way. The early computers did not do this because they had various technical limitations and trade-offs where getting a particular resolution or color palette on the screen was more important than the exact shape of the pixels.
You may still occasionally stumble upon special-purpose displays (think of something like a jumbo LED ad display on the outside wall of a shopping mall, or the LED array displays showing the next stop on a local bus, or the monochrome LCD display on the control panel of some industrial device) where the picture elements are not necessarily square-shaped, and where your pixel-graphics designs need to be scaled or shaped accordingly. That is, if you want to maintain the correct (physical) aspect ratio for graphics you output.
The less resolution and colors a display has, the more it calls for hand-tweaking your graphics pixel-by-pixel, or designing them from scratch for a particular graphics mode or display. Even more so if the final picture elements are not square. (Mere mechanical application of interpolation algorithms will usually produce quite bad results if the target resolution or color depth is small enough. Or inversely, the quality of your designs can be considerably better if you design for the limitations of the device and control the output to the level of the individual picture elements instead of just applying scaling algorithms and automatic conversions.)
The need for these considerations is altogether getting rarer now as even the lowest-end devices often have plenty of resolution and color on their displays, and engineers mostly try to make the addressable picture elements square in their shape, if at all possible. If you work with SD video (for archival or editing purposes), or design graphics for retrocomputing or demoscene projects, they’re still very real, though.