What were the three main purposes for which Leonardo da Vinci used drawing? What is hatching? What is crosshatching?

Leonardo da Vinci drew for several reasons, the most pragmatic of which was to make a living. Leonardo was born out of wedlock and was apprenticed at the age of fourteen to Verrocchio as a studio boy or Garzone since his artistic talent was evident at a young age. By twenty, he was certified in the Guild of Saint Luke's and his father set him up with his own studio. To move out of his father's house, he depended on commissions to support himself. The act of drawing was an important first step to creating paintings, a study or a first draft. While this was by far the most mundane and necessary reason to create art, Leonardo had other imperatives.

To keep at the top of his field, Leonardo used drawing to understand anatomy. At that point in time, many people based their own art on the art of those considered to be masters. Leonardo was trained from an early age by Verrochio that the best way to create realistic artworks, especially those of anatomy, was to actually study anatomy. As an artist, he had permission to dissect human corpses, and he availed himself of that, writing over 13,000 words on anatomy and making detailed sketches. He took it a bit further, dissecting many animals and comparing them to human anatomy.  There's an apocryphal story that after Leonardo started dissecting birds in order to create more realistic ones in his paintings, he urged others to do the same. However, they all began to copy Leonardo's birds instead!

Leonardo used hatching and cross-hatching in his drawings. Hatching is the usage of short parallel lines to indicate shading. Cross-hatching is the same, only in the opposite direction so that it creates a series of Xs or squares to show shading. 

The third main use Leonardo had for drawing was in his pursuit of engineering. He found employment as an engineer in Venice in 1499. He was fascinated by flight, trying for much of his life to make a flying machine. He is famously quoted as saying,

"For once you have tasted flight you will walk the earth with your eyes turned skywards, for there you have been and there you will long to return."

While his flying machines may not have been a success, he created a series of mobile walls for Venice in order to protect its walls. He also created a schematic to divert the Arno river.

Though many of his inventions were impractical, Leonardo was fascinated by the inner workings of things, be they machine or man, and used his drawings to support himself throughout his life.