About each category:

White Tea

White tea comes from the buds and leaves of the Chinese Camellia sinensis plant. The leaves and buds are allowed to wither in natural sunlight before they are lightly processed to prevent oxidation or further tea processing.

The name "white tea" derives from the fine silvery-white hairs on the unopened buds of the tea plant, which gives the plant a whitish appearance. The beverage itself is not white or colorless but pale yellow.

Green Tea

Green tea is made from the leaves from Camellia sinensis that have undergone minimal oxidation during processing. Green tea originated in China, but it has become associated with many cultures throughout Asia. 

Green tea has recently become relatively widespread in the West, where black tea has been the traditionally consumed tea. Many varieties of green tea have been created in the countries where it is grown.

Oolong

Oolong is a traditional Chinese tea (Camellia sinensis) produced through a unique process including withering the plant under the strong sun and oxidation before curling and twisting. Most oolong teas, especially those of fine quality, involve unique tea plant cultivars that are exclusively used for particular varieties.

The taste of oolong varies widely among different subvarieties.

Black Tea

Black tea is a type of tea that is more oxidized than oolong, green and white teas. Black tea is generally stronger in flavor than the less oxidized teas.

While green tea usually loses its flavor within a year, black tea retains its flavour for several years. For this reason, it has long been an article of trade, and compressed bricks of black tea even served as a form of de facto currency in Mongolia, Tibet and Siberia into the 19th century.

Pu-erh

Pu-erh is a variety of fermented dark tea produced in Yunnan province, China. Fermentation is a tea production style in which the tea leaves undergo microbial fermentation and oxidation after they are dried and rolled.

Pu'er traditionally begins as a raw product known as "rough" Mao Cha (毛茶) and can be sold in this form or pressed into a number of shapes and sold as "raw" Sheng Cha (生茶). 

Yerba Mate

Mate sometimes spelled maté in English, also known as yerba mate, chimarrão, is a traditional South American caffeine-rich infused drink, particularly in Argentina.

It is prepared by steeping dried leaves of yerba mate (Ilex paraguariensis, known in Portuguese as erva-mate) in hot water. It is technically not "tea" since it does not contain leaves of the camellia sinensis plant, but rather an "infusion."

Matcha

Matcha is a finely milled or fine powder green tea. The Japanese tea ceremony centers on the preparation, serving, and drinking of matcha.

In modern times, matcha has also come to be used to flavour and dye foods such as mochi and soba noodles, green tea ice cream and a variety of wagashi (Japanese confectionery). Matcha is a fine-ground, powdered, high-quality green tea and not the same as konacha.

via WikiTea

Rooibos

Rooibos, meaning "red bush" is a broom-like member of the legume family of plants growing in South Africa.

The leaves are used to make a herbal tea called Rooibos or bush tea (esp. Southern Africa).

In South Africa, it is common to prepare rooibos tea in the same manner as black tea and add milk and sugar to taste. Other methods include a slice of lemon and using honey instead of sugar to sweeten.

Herbal Infusions

Herbal tea, or tisane, is any beverage made from the infusion or decoction of herbs, spices, or other plant material in hot water, and usually does not contain caffeine.

These drinks are distinguished from caffeinated true teas (black, green, white, yellow, oolong, etc., which are prepared from the cured leaves of the tea plant, Camellia sinensis), as well as from decaffeinated tea, in which the caffeine has been removed.