I noticed the look of shiny grease on the bottom of my car so I took the bottom cover off and discovered this. It looks like it is coming from the place circled in the picture. If you know what this part or any advice would be appreciated.

 

The part in the picture is your passenger side sway bar bracket. Usually it is greased from the factory, but not to where you would get what you are seeing. It looks as if the leak (oil not grease) is coming from elsewhere & just accumulating at the point.

 

Mazmart is exactly correct. That bracket is not the source of your leak. If you raise the car and put a work light underneath, perhaps you'll find the source. Good luck...you'll find it!

David