Immigration: social sciences and humanities: U.S. Immigration Statistics

Compiled by the Urban Institute, provides data for each state and metro, showing the number of children of immigrants, how many are in low-income families, and where immigrant parents were born.
This article about the tool is also very helpful: including links out to additional sources with nicely-worded caveats about comparing data.

Annual core reports on immigrants by legal status and quantitative studies and research on immigration. See also: Yearbook of Immigration Statistics.
If the site goes down, or statistics become unavailable, you can find the archived site via the Internet Archive's Wayback machine, including pdfs and excel spreadsheets.

The U.S. Census Bureau uses the term foreign born to refer to anyone who is not a U.S. citizen at birth. This includes naturalized U.S. citizens, lawful permanent residents (immigrants), temporary migrants (such as foreign students), humanitarian migrants (such as refugees), and people illegally present in the United States.
The Bureau provides a variety of demographic, social, economic, geographic, and housing information on the foreign-born population in the United States.

Includes data on immigration court proceedings, decisions, appeals, asylum cases, etc.