The “base” stands for the number of characters you use to represent a number.
Binary, which is essentially Base2 (although no one really calls it that) represents a number using two characters, zeroes and ones.
Base10, which is what we’re most used to, represents numbers using 10 characters (0–9).
Base16, better known as hexadecimal, uses 0123456789abcdef — 16 characters.
Base58 uses (yes, you guessed it) 58 characters. It makes use of all the alphanumeric characters (0–9, a-z, A-Z) except, and here’s the clincher, all the easily mistakable characters. Base 58 does away with zeroes (0), capital i’s (I) and o’s (O), lower case l’s and doesn’t use symbols such as +’s and /’s.
The purpose of Base58 is to represent large numbers in a short format while avoiding easily-misinterpreted characters.