I don't know about other English speaking countries like England or Australia, but here in Canada the census defines a first-generation Canadian as "foreign born people over age 15" (the census has a similar definition in the US but the age might be different).

Note though that as long as one parent is foreign-born, you will be considered 2nd generation no matter how long the trail goes back for your other great-great etc grandparents. So,as @hsu_ag-member said, even if one parent can trace lineage back to the Mayflower, but the other parent is from say England, the kids are considered 2nd generation.

So in my own case, even though both my grandmothers were 2nd generation, I myself am only 2nd generation because my grandmothers married foreign-born as did my mom. My kids too will only be 2nd generation because I married someone born in Europe.

Note too, it doesn't matter if the parent is Canadian or American, the key is foreign-born. So if a 4th G American marries an American born in France who came back to the US at age 3, the kids are only 2nd G according to the government. I suspect that in our global age, there will be an increasing number of 2nd G people.

@Preston: I'm not positive, but most Asians use 1st G to mean the ones who came to the country, their kids are 2nd G.