The Star Wars films are always referring to past events unseen in the films. Oftentimes, the details of those events are depicted in the Expanded Universe. Here's where you can find out more about some of the more notable references:

"He has just returned from a border dispute on Ansion." (AOTC)
Obi-Wan and Anakin's exploits on the volatile world of Ansion are depicted in the novel The Approaching Storm.

"I haven't felt you this tense since we fell into that nest of gundarks." (AOTC)
Obi-Wan and Anakin in the nest of gundarks is depicted in the junior novel Jedi Quest #7: The Moment of Truth.

"I was recruited by a man called Tyranus on one of the moons of Bogden." (AOTC)
Jango Fett's recruitment is detailed in the Bounty Hunter video game and the Open Season comic series.

"...that business on Cato Neimoidia doesn't doesn't count." (ROTS)
The mission to the Neimoidian purse-world is detailed in the Labyrinth of Evil novel.

"Well, the bounty hunter we ran into on Ord Mantell changed my mind." (ESB)
Han has had more than a couple of run-ins with bounty hunters on Ord Mantell. The one he is referring to in this case is most likely the events of the newspaper strips that were reprinted in Classic Star Wars, Vol. 1: In Deadly Pursuit. Ord Mantell encounters are also depicted in the RPG adventure Scoundrel's Luck, the audio story Rebel Mission to Ord Mantell, as well as another encounter later on in the newspaper strip.

"Your ship? Hey, remember, you lost her to me fair and square."
The fateful sabacc game where Han wins the Millennium Falcon is dramatized in Rebel Dawn (1998, Bantam Books), the final novel of the Han Solo Trilogy. Heir to the Empire (1991, Bantam Books) revealed that Han selected his freighter from a lot of ships belonging to Lando, but the actual winning of the Falcon was not shown until Rebel Dawn. An alternate version of this event was shown in "In the Beginning..." Star Wars Tales #11 (2002, Dark Horse Comics). It was also depicted in a Galaxies Trading Card and in Insider #90

"Many Bothans died to bring us this information." (RJ)
The Shadows of the Empire novel and comic depicts the deaths of these "many" Bothans. For the Kevin Rubio version of events, check out The Return of Tag and Bink: Special Edition which depicts the fate of Manuel "Manny" Bothenz.