There are several ways to have fun on illustrator - you draw, paint and create amazing things from simple to the most complex.
The Width tool in the tool bar is a great tool among the many interesting tools in this software.
So lets see what amazing patterns and motifs you can make!
Step 1: Works Only With Strokes
The width tool is unique, it only works with the stroke/outline of a shape, a pen tool shape/line, or of a line tool. The width tool is in the tool bar panel of illustrator.
Step 2: The Way to Do It!
First of all select anything that has a stroke, an outline or is a path with a stroke thickness. Draw/Make your shape or path. Once the shape is drawn, select the path and then choose the width tool option from the tool bar and simply tweak it the way you want.
As soon as you come over the selected path of the shape's stroke with the width tool, you simply click at any point on the path or stroke and while clicking, move your cursor away and toward the stroke or path. You will automatically see how the stroke changes. Once you have decided how the stroke looks like, you can now make a pattern!
P.S. I am using Adobe illustrator CS 6