A felony, as distinguished from a misdemeanor, can be any particularly serious crime that is defined by state law. Individual states generally have discretion as to what crimes they will classify as felonies or misdemeanors, but generally, felonies are crimes which carry a penalty of at least one year in prison, while misdemeanors usually carry a year or less in prison or jail.
A felony arrest is simply when the police take a person into custody, on suspicion that they have committed a crime. A felony arrest can take place before or after a felony charge has been issued.
A felony charge is the beginning of formal legal proceedings against the person accused of the felony: it is when a district attorney brings formal accusations against the defendant before the court.
The exact felony charging procedure differs between individual states, but there are generally two different procedures.
Grand Jury – Several states and the federal government use a grand jury. A group of ordinary citizens are picked for the grand jury, and then the prosecutor presents evidence against the accused. The grand jury decides whether or not the person can be charged with a crime.
Note that this is not a determination of guilt; it is only a determination that there is enough evidence to bring the defendant to trial. Under the U.S. Constitution, a defendant accused of any "capital or infamous crime" (basically, any felony) cannot be brought to trial unless charged by a grand jury.
Prosecutor’s Complaint – In states that do not use the grand jury system the prosecutor writes a document accusing the defendant of a crime (known as an "information" or "complaint"), and presents it to a judge. The court, to ensure that there is enough evidence to go to trial, usually holds a preliminary hearing, in which they examine the available evidence, and decide whether or not the defendant should be brought to trial.
If you are involved in a criminal proceeding that involves felony charges, you should immediately contact an experienced criminal law attorney. An attorney will help you understand your case and will represent your best interests in court.