Like most saints, there is no single specific cross associated with St. Nicholas, and in his case, we hear more of Santa Claus than Saint Nicholas himself or his cross.
Even so, he is one of the most popular saints, especially for Eastern Catholics and the Orthodox. Due to his charity and concern for the welfare of children, the Archbishop Nicholas (280-342 A.D.) of the Mediterranean port of Myra (Turkey) became Saint Nicholas and is the patron of children and travellers.
This eastern background is the reason the St. Nicholas Cross is usually an Eastern Orthodox style superimposed on a Budded Cross and often embellished with the acronym IC XC NIKA1.
For a little more on Santa Claus, see Pagan background of Christmas. See also the Christmas Natal Cross.