Several possibilities.

Original immigrant may have married a woman of color(black or native), so James is listed as Mullato
or, if James was white he may have been listed as a Mulatto because he was married to a non-white.

Do you know when he is supposed to have arrived from Ireland?

Where were James and his family in 1870? How old?
Give us some details so we can assist better.....

below is a link to check Irish Census records, but they are county by county.