The orbit of an element in the set the group is acting upon, S, is everywhere in S it is sent to by an element of the group. So, for example, there is no way of sending 1 to anything other than 2. Similary, g sends 1 to 2.

The stabiliser of an element is the set of all group elements which "stabilise" the element; group elements which don't send the element anywhere.

So, for example, but . Therefore, stabilises 1, as does the group identity. Therefore, the orbit of the element 1 is {1, 2} and its stabiliser is the subgroup . Note that 2.2=4=|G|.

Does that make sense?