Photoshop Elements is essentially a "Lite" or hobbyist version
of the regular Photoshop.
Lightroom and Photoshop (either regular or elements) do
different things and target different audiences. There is some
overlap in image editing functions, but the easiest way to describe
it is Lightroom is a complete photo and data management solution.
Photoshop is not.
So it really depends on who you are and what you want to do with
the product. Serious photographers generally purchase Lightroom,
whereas folks who want basic image editing might get Elements, and
for gold-standard do-all professional-level editing you get the
full Photoshop. Plenty of people get both Lightroom and
Photoshop.
For more in-depth details, see the Related Links.