What is the botanical name of yam
The botanical name of yam is Dioscorea spp.
This is so because there are different species and varieties of yam with each having their own botanical name.
Yam belongs to the kingdom plantae
It's Subkingdom  is that of a group of  Vascular plants called the subkingdom Tracheobionta.
It is a spermatophyta (seed bearing plant) 
It belongs to the division Magnoliophyta which are flowering plants
It is a monocot of the class Liliopsida and subclass Lilidae
It is a member of the Order Liliales and family Dioscoreaceae.
The Genus of yam is known as Genus Dioscorea
Please read Yam planting procedures and production techniques

Species/Varieties of yam and their botanical names

The white yam - Dioscorea rotundata

This is a widely grown species of yam in Africa with over 200 varieties cultivated.
White yam tuber is roughly cylindrical in shape, the skin is smooth and brown, and the flesh is usually white and firm.

Yellow yam - Dioscorea cayennensis

It is most time considered to be the same species as white yam.  Yellow yam has yellow flesh, caused by the presence of carotenoids. It looks similar to the white yam in outer appearance; its tuber skin is usually a bit firmer and less extensively grooved. The yellow yam has a longer period of vegetation and a shorter dormancy than white yam.

Water yam - Dioscorea alata

This is also called purple  or winged yam. The shape of its tuber  is generally cylindrical, but can vary. Tuber flesh is white and watery in texture.
It is the most cultivated species of yam in Asia and West Indies. It is the most distributed yam owing to its wide cultivation network.

Chinese yam - Dioscorea polystachya

This species of yam originates from China. The Chinese yam plant is kind of smaller than the African, with the vines about 3 m (10 ft) long. It is tolerant to frost and can be grown in much cooler conditions than other yams. It is also grown in Korea and Japan. It was later introduced to Europe.

Wild bitter yam - Dioscorea esculenta

This is the third most cultivated yam. Originally from southeast Asia. The tubers are eaten baked, boiled, or fried much like potatoes. Because of the small size of the tubers, mechanical cultivation is possible, which along with its easy preparation and good flavor, could help the lesser yam to become more popular in the future.

Bitter yam - Dioscorea dumetorum

 the bitter yam, is popular as a vegetable in parts of West Africa, in part because their cultivation requires less labor than other yams. The wild forms are very toxic and are sometimes used to poison animals when mixed with bait. It is believed that they have also been used for criminal purpose.

Cush-cush yam - Dioscorea trifida

This species of yam is native to the Guyana region of South America and is the most important cultivated New World yam. Since they originated in tropical rainforest conditions, their growth cycle is less related to seasonal changes than other yams. It is relatively easy to cultivate and have good flavor.