While you want to avoid committing any crimes at any point in your life in the United States, it's particularly wise to do so when you're either visiting the country on a visa or are merely a permanent resident.
Committing any criminal offense, whether it be a misdemeanor or felony, during one of these latter stages can result in you having your legal status downgraded or even get you deported. You may even be barred from entering the U.S. again.
Crimes that are considered to involve moral turpitude are those a judge may deem to adversely impact society's moral standards. Perjury, fraud, concealed weapons, tax evasion and child abuse charges are all considered to be moral turpitude offenses.
As far as aggravated felonies are concerned, crimes such as drug trafficking, murder and firearms charges were considered to be some of the only crimes that fit this bill up until 1988. Congress then added other crimes including theft, failing to appear in court, consensual sex between minors, battery and even tax fraud to the list of aggravated offenses.
What will become of your ability to remain here in the U.S. largely depends on your legal status here in the U.S. If you're in the U.S. on a nonimmigrant visa, then the the U.S. Citizenship and Immigration Service (USCIS) will generally seek to have you removed from the country the moment you rack up two misdemeanors or a single felony.
In contrast, any permanent resident (PR) who commits an aggravated felony may immediately be detained pending deportation. If a PR is ultimately removed from the country then reattemps entry, then he or she can be imprisoned for as long as 20 years for doing so. Even if his or her deportation is not ordered, then a PR may be restricted from ever applying for naturalized citizen status here in the U.S.
Asylees within PR status may only be deported if the crime they commit is deemed to be particularly serious. Refugees, however, can be depoted for most any felony, even if they may face significant danger back in their home countries.
If you've committed a crime in the U.S., whether here on a temporary visa or as a permanent resident, then a Montgomery County attorney can provide assistance in your legal matter.
Source: FindLaw, "How does a felony affect immigration status?," accessed March 02, 2018