Photoshop is an image editing software. It's for editing and manipulating images, which includes photos. With enough know-how, it can do just about anything you could want in terms of digital photos. It also has the capability to help you organize and archive your photos, mainly with the 'Birdge' side program. You can automate many things, including writing your own 'actions' or getting actions from other people. For many years it was the only software a photographer needed...and many today still use it primarily.
Lightroom is more of a 'workflow' software. It was designed to help photographers work with large numbers of images, quickly and efficiently. Since it was first introduced, there have been many improvements and it does have many 'image editing' tools. It is not a replacement for Photoshop, but many photographers find that they need to use Photoshop less often when using Lightroom as their primary workflow software.
Lightroom, also makes it easy to do a lot of basic things like saving a group of files for uploading to the web, or for printing etc. It will even create a slideshow or web gallery for you. Photoshop can do all of those things as well, but maybe not in such a streamlined way.
The 'big' difference between the two, which may not be appearant on the surface, is the way they work with files. Photoshop works the same way that most programs have always worked. It opens a file and then you have the option of saving that file again, or saving a copy if you don't want to overwrite the original. The changes you make to a file are applied to it by changing the pixels...sometimes called a 'destructive' workflow. You can work around this by using layers etc, but then you are limited to certain file types (TIFF, PDS etc) and those are very large files.
Lightroom, on the other hand, works off of a data base and is a non-destructive workflow. You 'input' the files, and LR takes note of the file location and saves a preview of it. While in LR, any change you make is recorded in a side-car file that is separate from the actual file. The original file is never actually touched so it's integrity is maintained and you don't even have to save multiple copies because the changes are stored in the side-car file.
When you are finished with the image in LR, you can choose to 'output' which saves separate files, giving you all the options for file type, size, compression, EXIF etc.
You can even integrate the two programs. From LR, you can choose to 'Edit in...' which takes the image into Photoshop so you can make specific edits that LR can't do. You can then save the image which takes you right back into LR, giving you the option replace the working image in LR or save a virtual copy.
So ideally, the combination of LR and PS is the way to go...but as for which one is better for you needs, it really depends on how you work and the type of editing you like to do.