Free members are the only ones who see ads. As far as I know, if you're a paying member (I think this applies for Gold as well (which is the cheapest membership)), they "hide" the ads for you.
But I know what you're talking about. When I was a free member, most of the ads that showed up were based on my recent activity. It can get annoying.
Has anyone noticed that chess.com determines what ads to display on its webpage based on your recent activity? For instance, I haven't played Runescape for about 8 months now, but one day, after I visited the Runescape website once, the next time I logged on to chess.com there was an ad about old-school Runescape returning, which I had never seen previously. And about a week ago, after I arrived in China for vacation, I immediately started getting ads in Chinese.
Of course, sometimes chess.com messes up. Right now I'm getting political ads related to the Nigerian elections, for some reason.