Like many law enforcement agencies, Immigration and Customs Enforcement (ICE) is perpetually underfunded and can't take action against everyone who has violated the law. Therefore, U.S. immigration law contains a concept known as prosecutorial discretion (PD).
Literally speaking, PD means that ICE has the power to stop working on someone's deportation (removal) case. By exercising PD, ICE can shift time and resources to high-priority cases that have implications for national security, border security, or public safety.
The existence of PD was sufficiently widely acknowledged that, at least in the past, ICE was at times willing to reverse course on an attempted removal and actually join the non-citizen in asking the immigration judge to terminate court proceedings and close the person's case. ICE has also been known to agree to ask an immigration judge to reopen a case so that the person could apply for relief from removal.
Under the Obama administration, ICE and other immigration agencies were directed to prioritize cases that were threats to national security, border security, and public safety (people who had committed felonies, attempted to cross the U.S. border unlawfully, terrorists), people who had committed misdemeanors and "new immigrants" (those who entered the U.S. prior to 2014), and people who had already been ordered removed (deported) from the United States. But President Obama also directed ICE and related agencies to offer PD to immigrants in removal proceedings who were not within these enforcement priorities.
With Donald Trump having taken office, however, and issued the "Presidential Executive Order: Enhancing Public Safety in the Interior of the United States," this picture has changed completely.
The short version is that nearly everyone is now considered an enforcement priority. Trump's order names as priorities for removal people who have been convicted, charged, or even committed acts constituting any criminal offense; who have engaged in fraud or willful misrepresentation in connection with any official or government matter; who have abused any public benefits program; or who otherwise, in the judgment of an immigration officer, pose a risk to public safety or national security.
Although the focus is on "crimes," notice that instead of actual convictions, mere criminal charges or suspicions by immigration officers will be sufficient to put someone at the top of the priority list. This list also sweeps in anyone who has used a fake green card or Social Security Number.
As if it weren't already clear that the notion of prosecutorial discretion has been dealt a mortal blow, a follow-up implementation memo by DHS Secretary Kelly states that "prosecutorial discretion shall not be exercised in a manner that exempts or excludes a specified class or category of aliens from enforcement of the immigration laws." In other words, there is no remaining directive as to who is at the bottom of the list and should be left alone.
The Kelly memo also states that "The exercise of prosecutorial discretion with regard to any alien who is subject to arrest, criminal prosecution, or removal in accordance with law shall be made on a case-by-case basis in consultation with the head of the field office component, where appropriate, of CBP, ICE, or USCIS that initiated or will initiate the enforcement action."
That suggests that no one whose case isn't uniquely compelling can expect to receive a grant of prosecutorial discretion in the future. That said, for someone who finds him- or herself in deportation (removal) proceedings, this may still be a remedy worth asking for. Consult with an immigration attorney for details.