Simply put, a crosier is a staff used by bishops and abbots. The Irish crosier was a little different than those used in mainland Europe. From Ask About Ireland:
"...crafted in Ireland from the eight or ninth century up until the end of the twelfth century. The origins of the Irish-crosier, like its continental counterpart, lie in the shepherd's crook. Both abbots and bishops carried them as insignia of their office and they signify the pastoral care of the congregation. However, Irish-crosiers demonstrate a divergence from the main tradition of crosiers in the Western Church. The significant difference is the distinctive shape of the head, which curves in a crook shape with the addition of a short pendant drop at its extremity. This is in contrast to the volute or simple walking stick shape of contemporary English and Continental examples."