After a little research on red headed birds of Texas, I found that he is a House Finch. I saw his mate a little while later and got a picture of her also. The female did not have any red feathers.
@paulchantler Paul, I was thrilled to see this little guy and his mate. The grackles and mockingbirds are running all the other birds away, it seems. The mockingbirds were picking on this one and his mate.