Lightroom is an independent product with a different purpose. Lightroom is a non-destructive digital darkroom product for making limited changes to images coming from a camera. It is particularly good at working with RAW files and has built in RAW handling. Since it is non-destructive, it does not save the image with changes applied, but rather saves a list of changes that need to be applied to an image.

While it supports integration with Photoshop (if you have Photoshop) so that you can do more advanced work either with the Lightroom image being used as a Smart Object, or by rasterizing (making the changes from Lightroom permanent on a copy) the image. It is not necessary to have Photoshop unless you want to use that integration.

Lightroom is also perfectly capable of exporting the images with adjustments applied permanently as a final output from Lightroom.