The three tools approximate three general levels of functionality and capability. Photoshop Elements is an entry-level image editing package. It provides image editing and some library management capabilities. It has some rich capabilities, and supports all the common editing capabilities of Photoshop. Adjustment layers allow non-destructive editing to a degree, however there are fewer adjustments possible in Elements than Photoshop. It is limited, and if you do a lot of photographic editing, you will eventually run into Elements' many limitations.
Photoshop Lightroom is a professional image editing package. It is specifically geared towards the digital photographer with an advanced workflow, RAW image support, and extensive library management capabilities. It is designed to expose the key tools important to editing digital photographs, such as a histogram, white balance and tone adjustments, color adjustments, detail adjustments (i.e. noise reduction and sharpening), as well as some advanced features like camera and lens profiles to correct distortion and vignetting. Lightroom also offers some rich tools that go beyond the basic on-screen editing, including print and book creation, slideshow generation, and web site publishing. New publishing modules in Lightroom 3 allow direct upload of your images and metadata to sites like Flickr and Facebook.
Photoshop CS is the ultimate image editing program. It is a low-level tool, giving you direct and total control over pretty much any edit you can think of. It's capabilities extend far beyond photographic editing and support artistic image creation and editing of a very wide variety. For a photographer, Photoshop brings HDR generation and editing, panoramic stitching, and a whole host of additional tools that are not available in Lightroom. Photoshop lacks some things provided by Lightroom, however, particularly its ease of use and clean workflow designed specifically to optimize a photographers image management, editing, and publishing. The most common and useful operations that need to be performed on digital photographs can be completed in much less time with Lightroom. Photoshop itself lacks any real kind of library management outside of Bridge.
If you do very light weight photography with only the most basic of edits, use Photoshop Elements.
If you want more control and/or have a large catalog of images (more than a few thousand), or want Photoshop-level support of RAW tools, Lightroom is highly recommended. It can act as the central tool for managing, editing, and publishing your photographic works. It has an ideal price point, making it easy to acquire.
Photoshop CS is a great tool to have to cover the cases where Lightroom is lacking, such as pixel-level editing or modification of your image. In previous versions Lightroom lacked the ability to soft-proof (simulate printed output on the screen) and support for ICM color profiles for various output devices, but this is no longer true.