If either target is illegal at the time the ability resolves (whether because it has left the battlefield, because it has gained Shroud or Protection, or for any other reason), the ability taps the remaining target, but no damage is dealt. If both targets are illegal, the ability doesn't resolve.

The power of the creatures when the effect resolves is used to determine how much damage is done.

In multiplayer games, you can choose a different opposing player each time it the ability is activated.

Either or both creatures can be tapped prior to using the ability and it still works.

Arena : règles, fonctionnement et interactions (MTG)

Arena is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

, : Tap target creature you control and target creature of an opponent's choice they control. Those creatures fight each other. (Each deals damage equal to its power to the other.)

How it works : Arena

Original English text — Wizards of the Coast / Scryfall

#1September 25, 2006· Wizards of the Coast
If either target is illegal at the time the ability resolves (whether because it has left the battlefield, because it has gained Shroud or Protection, or for any other reason), the ability taps the remaining target, but no damage is dealt. If both targets are illegal, the ability doesn't resolve.
No other card is named in this ruling.
#2October 4, 2004· Wizards of the Coast
The power of the creatures when the effect resolves is used to determine how much damage is done.
No other card is named in this ruling.
#3October 4, 2004· Wizards of the Coast
In multiplayer games, you can choose a different opposing player each time it the ability is activated.
No other card is named in this ruling.
#4October 4, 2004· Wizards of the Coast
Either or both creatures can be tapped prior to using the ability and it still works.
No other card is named in this ruling.

Interactions with other cards

These cards' text mentions Arena.

Undercity // The Initiative

You can't enter this dungeon unless you "venture into Undercity." Secret Entrance — Search your library for a basic land card, reveal it, put it into your hand, then shuffle. (Leads to: Forge, Lost Well) Forge — Put two +1/+1 counters on target creature. (Leads to: Trap!, Arena) Lost Well — Scry 2. (Leads to: Arena, Stash) Trap! — Target player loses 5 life. (Leads to: Archives) Arena — Goad target creature. (Leads to: Archives, Catacombs) Stash — Create a Treasure token. (Leads to: Catacombs) Archives — Draw a card. (Leads to: Throne of the Dead Three) Catacombs — Create a 4/1 black Skeleton creature token with menace. (Leads to: Throne of the Dead Three) Throne of the Dead Three — Reveal the top ten cards of your library. Put a creature card from among them onto the battlefield with three +1/+1 counters on it. It gains hexproof until your next turn. Then shuffle. Whenever one or more creatures a player controls deal combat damage to you, that player takes the initiative. Whenever you take the initiative and at the beginning of your upkeep, venture into Undercity. (If you're in a dungeon, advance to the next room. If you're not, enter Undercity. You can take the initiative even if you already have it.)