There's apparently a law from, oh, 1996 or so that allows outside groups to campaign for whoever they like, without being counted against "their" candidate's budget restrictions. The idea is that if they're just people like you, they should get to talk about it all they like, no matter how much it costs. There's no good way to keep people from expressing their opinions anyway (short of police action -- but that's more punitive than preventative) so you might as well not limit them money-wise. This time around, the main presidential campaigns are getting help from these groups -- donors are sending money to them (rather than / above-and-beyond sending money to the main campaigns themselves) knowing they can make big contributions with a big effect.
So, now Kerry has a problem with one of these third-party ads, and is asking Bush to do something about it (publicly saying "stop those ads", even.) Why would Bush have any control over that? Why even ask him, except to try to pin any hard feelings voters might have about those ads on Bush? It's freedom of expression at work -- if it's false, you can do something about it (slander/libel are more than frowned upon) directly. Then again, you can't easily undo the damage done by false news articles or ads, particularly not in a timely manner when vying for an elected position.
I have a feeling these ads will start to get "smart" -- perhaps going so far as campaigning for one candidate by doing a really bad job of campaigning for her opponent(s), or being overly enthusiastic (in a sort of too-good-to-be-true way) about a candidate to the point of convincing voters to mistrust him. They can easily make a candidate seem useless by decreasing the signal-to-noise ratio, creating ads for a candidate containing nothing but non-issues like "George Bush wants to make sure you have free access to the best sexual-enhancement technologies". No matter how ugly it gets, the candidates really can't do anything about it except pin the ugliness on each other or happily accept the benefits of the ads. Besides, that's always more fun that talking about your position on big and small issues alike.
Update: I hear Bush said "all the ads" should stop -- every last third-party ad. I find that interesting, in that it goes above what Kerry asked for, and into new territory: Kerry asked him to ask the people making an ad he considered unfair to stop -- Bush instead asked them all to stop, without mentioning the particular one Kerry found unfair. Now, although Star Wars fans will remember the trash-compactor scene fondly, with C3PO saying something along the lines of "no, shut them all down" to R2D2, in an effort to make sure it got done correctly ... there's a chance this goes into the land of "free speech" and what-not, which maybe Bush shouldn't do so hastily. Then again, there's some neat stuff in the concept of people dueling it out with no outside interference -- except this whole presidential election (emphasis on election) is about outsiders determining the outcome. So that's a little screwed. Do you want to be able to speak commercially in favor of candidates if you so choose? Do you want others to be able to do so against your favorite candidate? Or do you just want people not to be jack-asses about it?