Thailand enforces a 500 Baht/day fine for overstaying a visa. Although this is technically due for each day that you overstay including the first, it is often waived for overstaying only a single day. If you were to overstay by 2 days you can guarantee that you'd be hit with a 1000 baht fine, and so on up to a maximum fine of 20,000 baht.
They also stamp your passport with an "overstay" stamp to record the fact that you overstayed. Whilst this will not block you from entering the country in the future, if you are found to overstay a second time you can expect additional action to be taken. Again it's likely that they will not stamp your passport for a 1 day overstay.
Even though this fine seems relatively low (for a few days at least), it is NEVER a good idea to overstay a visa. The fine applies if you are "caught" at the airport when departing the country. If you are caught elsewhere in Thailand you can expect to be jailed for at least a day or two, and then deported from the country. Again this is not likely to be an issue for a single day overstay, but for a longer overstay it could be!