What is this?

Thank you to everyone who suggested ways for me to display my buttons.  One of the ideas I got last week about what to do with my button collection came from Cindy.  She separated hers by color and stores them in jars, for now that's what I plan to do.  I'm not doing the whole collection at once, just starting with the shoebox that I dumped out on the dining room table for last Thursday's blog.   I sorted those buttons in the evenings as I watched TV.  It took a few nights to go through and as I got to the last of the buttons I found this:

It was much dirtier than this - I couldn't even tell if the center hole was round or square.
(I cleaned it a bit because in the first pictures you couldn't see any markings.)

I decided to wash it because I couldn't tell what it was.  I even used a little silver polish on it.  It took a bit of scrubbing before I could finally see markings.

Texas Quarter and ?

I love the saying "Sight is an ability, seeing is an art."  I can see this, but what am I looking at? 

And the flip side...

Now I'm curious.  What is it?  How do I go about finding out what it is?  Thinking of it as a coin will be my starting place, but I've never researched coins before.  Any tips on how to find out what I've got or does anyone know what this is?

Thanks,

cJoy

It's Thursday, so I'm taking my 'whatever it is' over to Vintage Thingie Thursday with Colorado Lady. Just use her button to join us.