A bhikkhuni (in Pali) or bhikshuni (in Sanskrit) is a Buddhist nun; the male version is bhikkhu (bhikshu in Sanskrit). Like bhikkhu, bhikkuni live according to the Vinaya, an exhaustive list of monastic rules, though the bhikkhuni's list of rules is longer. The extra rules subordinate bhikkhuni to bhikkhu, requiring, for example, senior bhikkhuni to pay homage to junior bhikkhu, a reversal of the standard hierarachy of seniority in the sangha. This in spite of the fact that the Buddha declared that men and women have equal spiritual potential.
Discriminatory or not, however, for many Buddhist women these rules can only be abstract, for many Buddhist traditions have not retained an order of bhikkhuni. Under the rather strict rules of ordination, a new order can only be founded when new bhikkhu or bhikkhuni are ordained by members of an existing order. As a Thai bhikkhuni lineage was never started from the original Indian order, there is no order of bhikkhuni in Thailand today.
When the Thai bhikkhu sangha has fallen into disarray - after periods of war and upheaval, for example - the bhikkhu have been "restored" by bringing bhikkhu from Ceylon, now Sri Lanka, to oversee ordination. The Thai sangha does the same when the Ceylonese lineage is destroyed. The Ceylonese bhikkhuni sangha, the first outside of India, did seed a bhikkhuni lineage in China, and though the Ceylonese line has died out, the Chinese one continues. Some rebel Thai women have been ordained in the Chinese lineage, though many Theravadins consider the Chinese lineage to be Mahayanist, not Theravadin. Still, for these women, it is their only way to lead a fully committed Buddhist life.
Travellers to Thailand may have seen women with white robes and shaven heads and thought them female counterparts to the bhikkhu with their orange robes and shaven heads. These women are not bhikkhuni, however; they are known in Thai as mae chi, and are lay Buddhist women who have chosen to leave the social world and pursue a path of heightened virtue, following five or eight precepts. They are not - indeed, cannot be - ordained as bhikkhuni in Thailand. Some mae chi live in wat (temples) with the bhikkhu, where they are apt to be treated as their servants and expected to spend their days cooking and cleaning for the bhikkhu instead of studying, and so some mae chi live in their own communities.
An excellent introduction to women and Buddhism - not just in Thailand - can be found in Chatsumarn Kabilsingh's little book, Thai Women in Buddhism, published by the Buddhist Parallax Press in California. Chatsumarn is a professor of religion and philosophy at the prestigious Thammasat University in Bangkok, and her mother is a rebel bhikkhuni who has received ordination from that lineage that in China by Ceylonese bhikkhuni.