Gauge is how many stitches per inch you should have in your knitting in order to end up with your end product being the same size called for in the pattern.
In this case, when you knit this, 13 stitches across should measure 4 inches, and 18 rows should measure 4 inches.
People knit at different tensions, so the size of the needle specified in the pattern does not always match what a knitter might need.
For garments, getting the correct gauge is important so the item will fit.
For things that don't need to fit, gauge is less important, as long as it's ok that your finished product is a different size than the measurements in the pattern say.
is an article about knitting a swatch and how to measure gauge.