Gauge is basically how many stitches per inch you knit.
A pattern gives a gauge so that your rendition of the pattern will turn out to be the same size as the measurements they give you.
People don't knit at the same tension, so gauge is given in a pattern. You do a swatch--a sample about 5 x 5 inches in and measure how many stitches you get across 4 inches. If you have too many stitches per inch, you try it on a larger size needle. If you have too few stitches per inch, try a smaller needle.
To get an idea of how important gauge is for a garment that needs to fit, think of it this way.
If you knit 100 stitches at 4 stitches per inch, your piece will be 25 inches wide.
If you knit 100 stitches at 5 stitches per inch, your piece will be 20 inches wide.
A big difference for one lousy stitch per inch, but they do add up.