An Authorization to Return to Canada (ARC) applies to someone  who has been removed from Canada and now wishes to return to Canada.

You cannot apply for ARC within Canada or at a port of entry.  An ARC must be submitted at the consulate or Embassy serving the foreign national.
Not all kind of removals require an ARC. There are three types of removal orders:

• Departure Order
• Exclusion Order
• Deportation Order

If you received a Departure Order, you must leave Canada within 30 days and while leaving Canada you must not forget to obtain a departure record confirming that you have left Canada within 30 days.

If you have received an Exclusion Order, you will not be allowed to come to Canada for 1 to 2 years. An ARC is will be required if you wish  to return prior to the expiration of the applicable period.

If you have received a Deportation Order, an ARC will be required under any circumstance. The vast majority of deportation orders are failed refugee claims. The typical example is a denied refugee claimant who leaves Canada, enters the US, marries a US citizen, becomes a permanent resident of the United States, then a US citizen, and then wants to return to Canada to visit family. In this case, an ARC is required even if the deportation occurred decades ago.

Other deportation orders stem from failed admissibility hearings or persons who were otherwise deported as a result of committing a crime.