Azadirachta indica - That's the botanical name for neem. Indica gives away the tree's origin, India.
The whole name, though of course latinized, comes from the Persian language Farsi:
So the botanical name translates into: "The Free Tree Of India"
Another common name for neem is Margosa.
Neem belongs to the botanical family Meliaceae. That means it is related to the mahogany tree. (The Meliaceae family is also called the mahogany family)

The alternate, pinnate leaves are 20-40 cm (8 to 16 in.) long, with 20 to 31 medium to dark green leaflets about 3-8 cm (1 to 3 in.) long. The terminal leaflet is often missing. The petioles are short. Very young leaves are reddish to purplish in colour. The shape of mature leaflets is more or less asymmetric and their margins are dentate with the exception of the base of their basiscopal half, which is normally very strongly reduced and cuneate or wedge-shaped. Razor Leaf.
The (white and fragrant) flowers are arranged axillary, normally in more-or-less drooping panicles which are up to 25 cm (10 in.)long. The inflorescences, which branch up to the third degree, bear from 150 to 250 flowers. An individual flower is 5-6 mm long and 8-11 mm wide. Protandrous, bisexual flowers and male flowers exist on the same individual.
The fruit is a smooth (glabrous) olive-like drupe which varies in shape from elongate oval to nearly roundish, and when ripe are 1.4-2.8 x 1.0-1.5 cm. The fruit skin (exocarp) is thin and the bitter-sweet pulp (mesocarp) is yellowish-white and very fibrous. The mesocarp is 0.3-0.5 cm thick. The white, hard inner shell (endocarp) of the fruit encloses one, rarely two or three, elongated seeds (kernels) having a brown seed coat.