Most of the time, in the U.S., they are talking about people who crossed the border illegally--in fact, mostly they're not even talking about all the groups who cross the border illegally, but specifically the Hispanic group. There are a variety of reasons for this, ranging from xenophobia through to larger cultural isolation making the community more visible as its own thing. They are, for the most part, undocumented, have entered illegally, and their continued stay is unauthorized. So any of "undocumented", "illegal", or "unauthorized" applies. Illegal clearly increases stigma, undocumented avoids it entirely (despite the illegal entry), and unauthorized is simply accurate.

The next most common usage is everyone, which includes the documented unauthorized folks including whoever has overstayed their visa. "Undocumented" is wrong for many of these folks, "illegal" is arguably true in almost all cases, but again, unauthorized is simply accurate.