Let's take a look at changing colors in Tunisian Crochet. It is very easy, and you can choose whether or not you want to change on the left side of the work or on the right.
Each way gives a different look. In the picture below, you can see that when changing colors on the left side of the work, the vertical bars of the previous color overlap the new color. That is not so when the color change happens on the right side of the work.
To change colors on the left side of the work:
Complete a forward pass. Drop the old color and, with the new color, work a return pass as normal. Draw the new yarn through one loop, then yarn over and pull through two loops for the rest of the row.
To change colors on the right side of the work:
Work your return pass until there are two loops left on your hook. Drop the old color, and with the new color, pull through the last two loops. If you are working in simple stitch as I did in the sample below, continue as normal; if you are working from a pattern or particular stitch, work as your instructions indicate.
That's it. Below is a short video showing the color changes.