In 'Pirates of the Caribbean 2' it is said Jack somehow 'marked' Beckett with something. What exactly did he do?

Beckett offered Jack Sparrow 'Letters of Marque' (his freedom) in return for his compass, that points to the heart's desire.
Note that Marque is in the same family as Marquee, and comes from the French "marquer", meaning "to mark". Jack
https://en.wikipedia.org/wiki/Captain_Jack_Sparrow

I personally was under the impression in this movie that Jack and Beckett had had a run in before (as it zoomed in on the burn on Jack's arm, which had "marked" him as a pirate,) but I'm not exactly sure what Jack "marked" Beckett with. It is possible that we never saw this mark of Beckett.

This is purely from my own knowledge (take note that I am a hardened POTC fan,) and that I have not exactly looked this up, and so may be wrong. But this is what seems logical to me, as we saw Jack's "P" which had been burned into his arm by the East India Trading Company, which marks him as a pirate.

Yes, what thegogga said is correct. I misunderstood what you were trying to ask. I am also a huge fan, and I remember very clearly the scene now that you have reminded me.

https://en.wikipedia.org/wiki/Cutler_Beckett

Here it says that Cutler Beckett never revealed the mark that Jack Sparrow left on him, and confirms that the mark that Beckett left on Jack Sparrow was a branding (the P mark).

Therefore we can only guess what Jack's mark was... I would guess it's something personal, or deeper than skin, due to Beckett's slightly wistful air when he admits to Jack's mark - maybe Jack stole something personal, or hurt someone precious to Beckett? Maybe Beckett and Jack are some way related / connected? Nothing is sure.

The Letters of Marque is just a prononciation coincidence... Oh well =P.

Trouble325