To do fast multiplication of double-digit numbers, you only need to understand addition and multiplication by single digits. If you have a sharp mind, you can even use this fast method to multiply double-digit numbers in your head. If you need to see what's going on, grab a pencil and paper and follow this simple algorithm.
Write down the two numbers that you're going to multiply. Call them AB and CD, for example, where each letter represents a digit. If you're able to recall the numbers in your head, you can skip this step.
Choose one of the numbers to be the main factor. It doesn't really matter which one you choose, so pick one at random. Assume that you've chosen AB.
Multiply AB by D and write this number down. Place a zero underneath the digit in the ones place of the result.
Perform the multiplication of AB by C and write this to the left of the zero you wrote in Step 3.
Add the two new numbers. This is the result of the multiplication.
If you want to make the process even simpler (although it might not be as fast), you can perform four single-digit multiplications and add the results. Multiply B by D, multiply A by D (and add a zero), multiply B by C (and add a zero) and multiply A by C (and add two zeros). Add these four numbers to get the answer. If you're able to do more of this process in your head, you'll get an answer faster.