While there are many books, kits and Certified Zentangle Teachers, known as CZTs, available to get you started in Zentangle, the method is simple enough that anyone can do it just by following a few simple structural rules. Zentangle is about the process, not the product.
Zentangle is a registered trademark for a very specific type of meditative drawing. Beyond the simple structure, there are no rules. Begin with a 3.5 x 3.5 inch square of high-quality paper. Textured paper is best, because it forces you to work more slowly and deliberately. Breathe deeply, relax, and maintain focus throughout the drawing.
First, draw a square border, about one-half inch from the edges. Use pencil, and draw lightly. Do not use a ruler; the roughness of the lines is part of the drawing. Within this border, and still using pencil and a light hand, draw one or more flowing lines or squiggles. These are called "strings," and serve to create structure. Then, using a fine black pen, preferably a Sakura Pigma Micro pen, fill in the spaces created by your string, with repeating patterns of lines or simple shapes, turning the paper while you are working. Finally, you may want to shade some areas using either pen or pencil.
Beyond these simple rules, there is no wrong way to do this, and no such thing as a mistake. The most important thing is that you draw deliberately, give the process your complete concentration, and enjoy the experience.