Hopefully this guide will help you understand binary and how a lot of the redstone circuits work. I will be adding more later on. Please post any questions, comments, or corrections below.
-= Part 1, Basics of Binary =-
The universal number system is a base-10 system, which means that we have 10 different number symbols. If you count from 0 up, you will count 10 numbers before increasing the number in the 10 place. Binary is a Base-2 system, which means that it has 2 different number symbols: 0 and 1. Counting in binary is no different from counting with normal numbers, when you get to that last number in that number place, you increase the number place to the left by one. Binary Counting
Adding in binary is not too hard, it's just like adding decimal numbers. Start with the column all the way to the right and add the 2 numbers together, if the total is 0 or 1, just write that down, if it's 2 then write a 0 and carry a 1 to the next column. If the number is 3 (which can be achieved through carrying), then write a 1 and carry a 1. Binary Adding