[MrDK]

You can print out a copy of this page from the Thai Embassy in London:
Tourist Visa Exemption

In accordance with the Immigration Bureau's Order number 608 / 2549 (2006), the above-mentioned passport holders will be permitted to stay in Thailand not exceeding 30 days and in total not exceeding 90 days within six months from the first date of entry.

Or this one from the Thai Embassy in the US which seems to explain it a bit better:
The Royal Thai Embassy, Washington, D.C.

Nationals from 36 designated countries may travel to Thailand for the purpose of tourism
as frequently as they wish without a visa and will be permitted to stay for a period of not
exceeding 30 days on each visit. They may, however, re-enter and stay in Thailand for a
cumulative duration of stay of not exceeding 90 days within any 6 month period from the date of
first entry.
The 36 countries include Australia, Austria, Bahrain, Brunei, Belgium, Canada, Denmark,
Finland, France, Germany, Greece, Iceland, Indonesia, Ireland, Israel, Italy, Japan, Kuwait,
Luxembourg, Malaysia, the Netherlands, New Zealand, Norway, Oman, the Philippines, Portugal,
Qatar, Singapore, South Africa, Spain, Sweden, Switzerland, Turkey, UAE, USA, and the UK.