Nerd Alert!
ASCII stands for American Standard Code for Information Exchange. ASCII characters are basically the same characters you use in real life. It has letters and numbers and punctuation and other special characters. So why the acronym?
Originally, computers were programmed in binary, which represented states of transistors: off (0) or on (1). As transistors got smaller and computers got faster, binary was built upon and replaced with hexadecimal. Hex is a base-16 numerical system, using characters 0-F. Each character actually translates down a level into binary (0 = 0000, 1 = 0001, 2 = 0010, ... , F = 1111). Well, as the programmers built upon these base systems, they decided it would be easier for most users to program in characters they could understand (a.k.a. English) so a system was devised to assign real characters into two-digit hex values. These "real" characters are called ASCII. So when you type an "A" on your keyboard, your computer sees "41" in hex.
As to why people call character drawings on their computers ASCII drawings, I have no idea. Like most computer terminology, my guess is it started with computer geek slang since most people don't look at what they're typing and think of it as ASCII.