We bought a area rug and put it down with a non slip mat underneath. Our dog pee'd on it over time, and it seems the urine combined with the mat stained the wood, in the pattern of the mat. It looks pretty ugly. I figure I'll have to sand and refinish the area. Anyone have advice for this specific problem? Not sure how to post a pic.

Urine stains the wood deeper than only the surface.You may be able to lighten the stain by sanding, but to remove stains you should replace with new flooring.Trying to bleach the stain is a waste of time. After sanding you may be able to diminish the stain look by coloring the floor with a dark pigmented stain.

I too am refinishing my old floors , and have a few pee stains from my dog , i used straight bleach on a spot thats going to be covered by carpet and it worked really good but did bleach the wood somewhat, i going to try baking soda and peroxide on the main floor have read it does pretty good job of it , i read about that online, then just sand and hope for the best, i know it wont all come out but if i can get most ill be happy.

If you have white oak flooring the staining does not penetrate as quickly as red oak. Deeper sanding in the stain area will help remove the stained wood.
Diagonal sanding with a coarse paper then straightening it off before going to a medium paper might take the stain off so you can blend the area with a light brown stain over the whole floor.

So would you say the answer is to refinish the entire floor and apply a darker stain? I do believe it is white oak.

You can stain a light brown, with some neutral stain over the dark area. After the neutral soaks in go over it with the brown color. The color that urine really knocks out are the reds, so you may want to use a colored pencil to help blend it in.I use #2 steel wool to really buff the stained area before staining and then after it is dried before using the colored pencil to help with the color.