Use "illegal alien", not "undocumented worker", "illegals", or "illegal immigrants"
What's the correct term to use for those who are here illegally? Here are some possibilities:
* "illegal immigrant": this term is a bit of an oxymoron. An "immigrant" is someone who's allowed under our laws to come to the U.S. to live, yet this term has someone do something they aren't allowed to do. So, this isn't the best. (But, "illegal immigration" is the shortest, easiest understood term for the process).
* "illegals": this term is a bit dehumanizing, and also sounds harsh. It's best to avoid using it.
* "undocumented", "undocumented immigrant", "undocumented worker", "unauthorized immigrant", or "unauthorized worker": these are mostly used as euphemisms by those who support illegal immigration (in many cases because they have some sort of financial or political interest in it). These terms might be technically correct in that illegal aliens probably don't have documentation. However, the usual intent of these terms is to deceive. Unless you have a financial or political stake in illegal immigration, avoid these terms. The forms that have "worker" in them are even more an attempt to mislead: not all those here illegally are working.
* "criminal alien": this term is specifically used to refer to illegal aliens who've committed some crime other than one relating to an immigration violation, such as drunk driving, capital crimes, etc. Calling all illegal aliens "criminal aliens" is inaccurate and gives supporters of illegal immigration an opportunity to point out that most illegal aliens don't engage in crimes not relating to immigration matters. Unless you want to help illegal immigration supporters, only use this term for those who've been convicted of non-immigration crimes.
* "invaders", "illegal invaders": there's a way to discuss the dangers inherent in having millions of foreign citizens inside your country, but using these terms isn't it. Only use these terms if you want to help illegal immigration supporters.
* "unauthorized alien": this term is used in the U.S. Code , but it's not in wide use otherwise so in other contexts it could be an attempt to mislead. Most people won't immediately catch on what this term means, so unless your goal is to mislead, avoid this too.
* "illegal alien": this is the best term to use. Almost everyone knows what it means, it's legally correct, and using it helps fight against the illegal immigration profiteers who'd prefer to use euphemisms.
The word alien has been used in the U.S. since 1798's Alien and Sedition Acts, and it's defined as "an individual who is not a U.S. citizen or U.S. national" . It doesn't imply that someone is bad, just that they aren't a citizen or national of the U.S.
The term "illegal alien" has been used several times in the U.S. Code (the main laws of the United States) and also in Supreme Court and lower courts' rulings, in Department of Justice and Department of Homeland Security press releases, and related government documents. See  for several examples.
An occasional complaint one will hear is that "illegal alien" is dehumanizing and that "no person is illegal". Yet, the adjective "illegal" is applied to some people who do things that are (not surprisingly) illegal and where the goal is not to dehumanize the perpetrator but simply to highlight that what they did was illegal.
Compare "no person is illegal" to searches for other terms using the "illegal" adjective (not all of these are relevant but not all about illegal aliens either; the number is the number of search results):
* "an illegal contractor" (73,600)
* "illegal racer" (28,200)
* "an illegal plumber" (2250)
* "illegal drug dealer" (1,910,000)
* "an illegal driver" (121,000)
* "illegal seller" (103,000; example here in the context of selling tickets)
* "an illegal dumper" (15,500)
For the latter, is the government of Louisville, Kentucky ("Report an Illegal Dumper Now", link) dehumanizing those who illegal dump waste? No, they're referring to a dumper whose actions are illegal. In the case of "an unlicensed plumber", is someone saying that the plumber as a person needs a license to exist? No, they're saying that the person - who might otherwise be a fine and upstanding person - is working as a plumber without a license.
When someone refers to an "illegal plumber" or an "illegal dumper", they aren't castigating that person any more than they would if they referred to "someone who illegally dumped toxic waste". They aren't saying that the dumper himself is illegal as a person, just that his actions as a dumper are illegal.
What illegal immigration profiteers would have us do is use euphemisms, and refer to that same person as an "unauthorized depositor" or similar.
Please use the correct term: illegal aliens. Don't fall for those who want to use euphemisms because they have a financial or political stake in illegal immigration.
 8 USC 1188, 8 USC 1255, 8 USC 1324, 8 USC 1324a, and 8 USC 1324b.
 The U.S. Code (the chief laws of the United States) uses "illegal alien" in several places:
* 2 USC § 658 - Definitions
...The term “Federal intergovernmental mandate” means... (II) the control of borders by the Federal Government; or reimbursement to State, local, or tribal governments for the net cost associated with illegal, deportable, and excludable aliens, including court-mandated expenses related to emergency health care, education or criminal justice; when such a reduction or elimination would result in increased net costs to State, local, or tribal governments in providing education or emergency health care to, or incarceration of, illegal aliens; except that this subclause shall not be in effect with respect to a State, local, or tribal government, to the extent that such government has not fully cooperated in the efforts of the Federal Government to locate, apprehend, and deport illegal aliens;
* 8 USC § 1252c - Authorizing State and local law enforcement officials to arrest and detain certain illegal aliens
* 8 USC § 1330 - Collection of penalties and expenses
The Secretary of the Treasury shall refund out of the Immigration Enforcement Account to any appropriation the amount paid out of such appropriation for expenses incurred by the Attorney General for activities that enhance enforcement of provisions of this subchapter. Such activities include... for the repair, maintenance, or construction on the United States border, in areas experiencing high levels of apprehensions of illegal aliens, of structures to deter illegal entry into the United States.
* 8 USC § 1356 - Disposition of moneys collected under the provisions of this subchapter
Such amounts as are deposited into the Fund shall remain available until expended and shall be refunded out of the Fund by the Secretary of the Treasury, at least on a quarterly basis, to the Attorney General for the following purposes- ...for expenses associated with the detention of illegal aliens.
* 8 USC § 1365 - Reimbursement of States for costs of incarcerating illegal aliens and certain Cuban nationals
* 8 USC § 1366 - Annual report on criminal aliens
the Attorney General shall submit to the Committees on the Judiciary of the House of Representatives and of the Senate a report detailing... the number of illegal aliens incarcerated in Federal and State prisons for having committed felonies, stating the number incarcerated for each type of offense;
* 8 USC § 1621 - Aliens who are not qualified aliens or nonimmigrants ineligible for State and local public benefits
(d) State authority to provide for eligibility of illegal aliens for State and local public benefits
* 42 USC § 6705 - Limitations on use of grants
(e) Performance of projects by State or local governments prohibited; competitive bidding; illegal aliens
* 49 USC § 40125 - Qualifications for public aircraft status
The term “governmental function” means an activity undertaken by a government, such as national defense, intelligence missions, firefighting, search and rescue, law enforcement (including transport of prisoners, detainees, and illegal aliens)
The term is also used in other government documents:
We affirmed the Board’s determination that the NLRA applied to undocumented workers, reasoning that the immigration laws “as presently written” expressed only a “ ‘peripheral concern’ ” with the employment of illegal aliens. 467 U.S., at 892 (quoting De Canas v. Bica, 424 U.S. 351, 360 (1976)). “For whatever reason,” Congress had not “made it a separate criminal offense” for employers to hire an illegal alien, or for an illegal alien “to accept employment after entering this country illegally.” Sure-Tan, supra, at 892—893. Therefore, we found “no reason to conclude that application of the NLRA to employment practices affecting such aliens would necessarily conflict with the terms of the INA.” 467 U.S., at 893.
* Another famous Supreme Court case, Plyler v. Doe (gave illegal alien children the same access to schools as citizen children) uses "illegal alien" throughout (link).
* The May 18, 2012 ICE press release "Houston-based company admits to hiring illegal aliens/Agrees to forfeit $2 million" (link).
* A DHS report (oig.dhs.gov/assets/Mgmt/OIG_06-33_Apr06.pdf):
This report assesses DHS’s Immigration and Customs Enforcement program for detaining and removing illegal aliens apprehended in the United States and at ports of entry.
Immigration and Customs Enforcement (ICE), an agency within the Department of Homeland Security, is responsible for identifying, apprehending, and removing illegal aliens... ...The Immigration Reform and Control Act of 1986 (IRCA), a comprehensive framework for “combating the employment of illegal aliens..." ...The state provision has three limitations: A detainee is presumed not to be an illegal alien if he or she provides a valid Arizona driver’s license or similar identification...
* Several proposed bills over the past several years; search at thomas.loc.gov