From Israel to Jordan

The rules somewhat differ between the 3 border crossings:

Jordan River (north):

Visa to Jordan – issued at the Jordanian terminal and costs 14 dinars.

Allenby / King Hussein (near Jericho):

Crossing the border at the terminal requires a valid passport with an expiration date at least six months after the departure date and all required documents for exiting:
Entry visa to Jordan clearly stating that the crossing is through the Allenby Border Terminal.

Yitzhak Rabin (Eilat - Aqaba):

Visa to Jordan:
Jordan requires a visa – for further information, please contact the Jordanian Embassy in Israel at 03-7517722.

So the easiest option is the Jordan River crossing.

On this website

How to get to Petra from Israel (Jerusalem, Tel Aviv & Eilat)

Note that visas must be arranged in advance and cannot be obtained at the border. Private cars and tour buses cannot cross – travelers must change vehicles upon crossing or use the other two border crossings.

We’d seriously recommend heading to the Jordanian Embassy in Tel Aviv to sort yourself out a tourist visa (ensure you have a passport sized pic). Once you have the visa, head to Allenby/King Hussein Bridge which is close to Jerusalem (about an hour away). You could always stop by in the Holy City to see some of Jerusalem’s attractions before heading to Jordan.

Ezra said in the comments:

If you are a citizen of Israel or Jewish it might get a little complicated. Just a warning. Once when on a trip to Israel, a man from the tour bus wanted to step over the Egyptian border and place both feet in either territories so he could say he'd stepped into Egypt, and even with a passport the guards wouldn't let him.