Speaking at a high school in New Mexico, Barack Obama was asked about immigration and, according to this liveblog of the event, said:
"We are a nation of immigrants," Obama answers.
"The only people can say that they aren't immigrants are the people sitting right here," the Presidential candidate says and points to the tribal leaders.
"There are some families who have been here for 4 or 500 years. They didn't cross the border, the border crossed them."
Certainly, he's somewhat factually correct: there are old Spanish origin families who are recipients of land grants from hundreds of years ago; they were there before New Mexico was part of the U.S.
However, he's coming at this from their side, which is in conflict with our - the U.S.'s - side. In fact, his quotes are straight out of Reconquista 101, and trying to find loopholes in and complaining about the Treaty of Guadalupe Hidalgo is a favorite pasttime of hundreds of minor extremists, with the TGH forming a major part of their lore. Sentiments like his are rarely expressed by those considered mainstream; the closest I'm aware of are Dolores Huerta and a LULAC official, and both of them aren't that mainstream at all.
As for the bit about Indians not being immigrants, one wonders at which point BHO would draw the line. Most of their ancestors came over thousands of years ago, but Mayflower descendents have ancestors who were here before 150 years before this was the U.S. and some European origin peoples' ancestors were here before then. Why is BHO differentiating between those who arrived from Europe and those who arrived from Asia? Why give so much weight to thousands versus hundreds of years? Assuming for the moment he wasn't just offering a dangerous pander, why not differentiate between those whose ancestors came here ten thousand years ago and those whose ancestors came here twenty thousand? Taken one step further, would Obama be like those who think that anyone with enough native blood - presumably more than one drop - can move anywhere within the Americas at will?
And, of course, the only people who are actually immigrants are those who are legal immigrants under our set of laws; those who were born here are not immigrants even if their ancestors were.
An open question is, other than pandering, what exactly BHO would have us do about all this? Should Mayflower descendents be considered among those who've had our borders cross over them? Is it good public policy to encourage U.S. citizens to think that our borders crossed over them? Would he give those who've been crossed over by our borders something? Would he support allowing anyone with the right amount of native blood to come here at will?
The rest of that section of his comments is his standard blather about this issue; the problems with that can be found here, here, and through a search. Discussing all the other reasons BHO's statements above are wrong is left as an exercise.
Note: I've confirmed BHO's statements with the author of the linked post.