Fairly accurately, I imagine. Statiscal sampling, if it's done properly, can be done highly accurately.
There are some potential problems. People with Nielsen boxes may cheat, and this could be done for different reasons. First, if the person likes a particular show, he may lie about how many people are watching the show. This could artificially boost ratings. Next, the person may change his viewing habits with the Nielsen box in the room. They may like a particular show, and although they don't feel like watching it (it's a repeat, a better, one time movie is on HBO, etc), they will either watch it (or at least leave the TV on so the box records it) or claim they watched it.
Political polling has a few problems: people don't want to admit they aren't voting for the minority race candidate, for instance.
What sampling can do, though, is allow you to ask a relatively small group of people, and extrapolate that to as if you asked everyone in the country. If there's a black guy running against a white guy, the white guy is usually going to get a few percent more than in the polls, but that's not because the polls were improperly sampled: if Gallup called and reached every household, there would still be the overstating of black support.
A problem that political pollsters have that Nielsen probably doesn't is determining if that person is really going to vote. Nielsen at least knows if you've turned on the TV.
Any poll also needs to make sure that the samples are representative of the population they are seeking to measure. Doing a nationwide poll of anything, but confining your sample to residents of Manhattan, is going to yield wacked out results. Oversampling Republicans or Democrats is going to skew the poll. In telephone polls, there's the risk that you are underpolling certain population segments not likely to take calls from strangers on their Caller IDs.
That is/was a problem with Nielsen's diaries. People write down shows that are 'popular' but they never watch it.
Next, the person may change his viewing habits with the Nielsen box in the room. They may like a particular show, and although they don't feel like watching it (it's a repeat, a better, one time movie is on HBO, etc), they will either watch it (or at least leave the TV on so the box records it) or claim they watched it.
That's possible, but Nielsen does meter every television set within a HH.