What is the fastest way to multiply two digit numbers?

The way I learned it:

$$\begin{array}{rrl} & 18 & \\ \times & 17 & \\ \hline & 156 & (10\times 10 + 8\times 7) \\ + & 70 & (10\times 7) \\ + & 80 & (10\times 8) \\ \hline = & 306 &\\ \hline \end{array}$$

In general:

$$\begin{array}{rrl} & {\rm AB} & \\ \times & {\rm CD} & \\ \hline & {\rm ACBD} & \text{(pad 0 if neccesary)} \\ + & {\rm BC}0 & \\ + & {\rm AD}0 & \\ \hline \end{array}$$

This is just dividing it into polynomial: $(10+7)\times (10+8)$. And it allows me to visualize easier than the traditional method.

The absolute fastest way to multiply two-digit numbers is to already know the result, by having learned the two-digit multiplication table just like you learned the one-digit multiplication table. Of course it means a lot of work up front.

My method for multiplying some two digit numbers is not going to work well in all cases, and has the draw back that you must memorize all the squares. But, I present it just as another trick you can learn. I +1'd FiniteA's answer as that seems like a nice way that works well in just about any case, and takes very little memorization or skill, i.e., it boils it down the simplest parts.

I would like to point out that my method is probably faster when the two numbers are relatively close together and when their difference is an even integer.

Using the fact that $(x - a)(x + a) = x^2 - a^2$, if you memorize perfect squares, you can do some multiplications pretty quickly.

Example 1: $24 \times 26 = (25 - 1)(25 + 1) = 25^2 - 1^2 = 625 - 1 = 624$

Example 2: $24 \times 27 = 24 + 24 \times 26 = 24 + 624 = 648$

Example 3: $24 \times 32 = 28^2 - 4^2 = 784 - 16 = 768$