From Outdoor Life (October) :

An exclusive interview with the two presidential candidates on gun rights, conservation and other issues that affect your hunting and fishing.

September 2004
[…]
OL: Are you a gun owner? If so, what is your favorite gun?

Bush: Yes. My favorite gun is a Weatherby, Athena 20-gauge (over/under).

Kerry: My favorite gun is the M-16 that saved my life and that of my crew in Vietnam. I don’t own one of those now, but one of my reminders of my service is a Communist Chinese assault rifle.

From the New York Times September 26 :

Senator John Kerry, a hunter who supported the recently expired assault weapons ban, frequently tells audiences he has never met anyone who wanted to use an AK-47 to shoot a deer. But it is not clear what Mr. Kerry does with the Chinese assault rifle he told Outdoor Life magazine he kept in his personal collection.
[…]
Andrew Arulanandam, a spokesman for the National Rifle Association - which has given Mr. Kerry “F” ratings throughout his career and backs Mr. Bush’s re-election - said the Outdoor Life comment made Mr. Kerry’s support of the assault weapons ban disingenuous.

”It’s O.K. for John Kerry to own these kinds of firearms, but it’s not O.K. for John Q. Public?” Mr. Arulanandam said, noting that if Mr. Kerry brought the gun home from the war as a souvenir he could be subject to court-martial. “He certainly owes people an explanation as to why there’s a double standard.”

Stephen P. Halbrook, a gun rights lawyer who has argued several cases before the Supreme Court, said the most common Chinese assault rifles, known as SKS clones, were not among the 19 models banned under the 1994 law. But some SKS’s have magazines holding more than 10 rounds, which violates a Massachusetts law against large-capacity weapons, Mr. Halbrook said. If the gun is fully automatic, Mr. Halbrook said, it is illegal in Massachusetts and would require a federal permit if Mr. Kerry kept it at one of his homes in Pennsylvania and Idaho.

Such permits are not public records.

Bob Ricker, a former N.R.A. lawyer who is now a consultant for the Coalition to Stop Gun Violence, said he was not worried by Mr. Kerry’s answer because “he knows a lot about firearms and he’s also one of the most credible individuals when it comes to talking about gun-violence prevention and what it takes to keep weapons of war off the street.”

From the New York Times of September 27 :

Senator John Kerry’s campaign said yesterday that Mr. Kerry did not own a Chinese assault rifle, as he was quoted as saying in Outdoor Life magazine, but a single-bolt-action military rifle, blaming aides who filled out the magazine’s questionnaire on his behalf for the error.

Michael Meehan, a spokesman for the campaign, said Mr. Kerry, the Democratic presidential nominee, owns two guns, a double-barreled 12-gauge shotgun and the rifle, which Mr. Meehan said Mr. Kerry “keeps as a relic” and had never fired. Mr. Meehan said the gun had no make or model markings on it and that Mr. Kerry “got it from a friend years ago,” adding that such rifles were first manufactured in Russia more than 100 years ago and were used by the North Koreans and the Vietcong.

The “Rifling” or grooves in a rifle barrel are there to impart spin. The latest, corrected version from the Kerry Campiagn makes it quite clear that it may have been from Vietnam (or even Korea or Cambodia), but was given to the Senator by a friend, so no crime was committed (by Kerry, anyway). And it wasn’t his fault, it was some staffer. And there are no markings on the weapon so it can’t be traced.
Given the “100 year old design” claim, it may be a Mosin-Nagant M1891 or one of its many variants, such as the Chinese type 55. Though most “M91”s have all sorts of serial numbers on various parts in unlikely places, e.g. at the magazine base….