A commonly-seen style of pistol used by many (though revolvers are also common). For examples from the crew's stash, Zoe weilds two during the raid on Adelei Niska's Sky Plex in War Stories, River shoots three guys with her eyes closed in the same episode, and lets not forget River's other experience with such a weapon in Objects In Space. River's "tree branch" was a chrome plated Desert Eagle from Magnum Research. With it's distinctive profile, it has cropped up in many movies including the Matrix movies as the Agents side-arm of choice.
In The Message, Mal and Zoe's old pal Tracey pulls an autoloading pistol on the crew and gives Wash a nice crease along the temple with it. He never gets the chance to fire it again.
(Special note: (The prop for that particular weapon is a favorite of Sci-Fi property masters - The Heckler and Koch VP70Z - a weapon of great historical importance. It was an early ultra high capacity pistol with an 18 round magazine, and was the first pistol to feature a polymer frame - introduced in the 1970's, it beat the more common Glock series by a decade. The military version, a VP70M with a slightly different profile, was featured in __ALIENS__ as the sidearm of the colonial marines.)