The referendum in Great Britain to leave the European Union was nicknamed "Brexit," a portmanteau of the words "Britain" and "exit." The same nickname was used in Germany. Let's take a moment to divert our attention from that event and focus instead on the different ways we can say "exit" in German.
Wir können die nächste Ausfahrt nehmen oder noch weiterfahr'n
We can take the next exit or keep on driving
Caption 7, Deutsche Bands - LuxuslärmPlay Caption
Die Ausfahrt is the most commonly seen translation of "exit" and is usually used in reference to the offramp of a roadway.
Wo ich reinkomm', geht man raus.
Where I enter, they exit.
Caption 43, Frank Zander - Hier kommt KurtPlay Caption
This is "exit" in its simple slang form of rausgehen, or "to go out."
Und zeigst nur stumm auf die Ausgangstür.
And just point silently to the exit door.
Caption 10, Herbert Grönemeyer - Was soll das?Play Caption
Der Ausgang is usually meant as the point where you leave a building.
Brüller [Umgangssprache], Lacher, Brüller, Abgang, mehr ist net [Dialekt, nicht] zu machen.
Big laugh, laugher, big laugh, exit, there isn't more to do.Play Caption
Der Abgang is the simple act of leaving or the act of leaving a specific sphere of activity. It can also mean the person who has left, as in "the one who left," although the more common terms are der Abgänger or die Abgängerin. Der Abgang is still occasionally used in medical and military circles to refer to death.
Das sind alles Leute, die eigentlich nichts weiter verbrochen haben,
als dass sie vielleicht einen Ausreiseantrag gestellt.
These are all people who actually violated nothing more than that they perhaps submitted an exit visa application.
Captions 27-28, DDR zum Anfassen - Ganz tief im WestenPlay Caption
Der Ausreiseantrag or "exit visa application" uses the word die Ausreise for "exit," which on its own would usually be translated as "departure." Hopefully the citizens of the United Kingdom —possibly soon consisting only of non-EU countries England and Wales — will not be required to file an Ausreiseantrag in order to leave what remains of the UK. It is nearly certain, however, that they will now have to get a residence permit (die Aufenthaltserlaubnis) to live in the EU.
Do a search on Yabla German to see the different ways that terms for "exit" such as der Abgang, die Ausfahrt, and der Ausgang are used in a real world context.