The gauge of a pattern is how many stitches per inch you need to knit in order to come up with the same sizes the pattern says. It is very important when you make a garment that needs to fit.
Everyone knits at a different tension, and if the designer of the pattern knits at 4 stitches per inch with a particular yarn and needle size, there's no guarantee that you'll get 4 stitches per inch with the same yarn and needles.
To get an idea of how much of a difference it makes, think of it this way. If a pattern has 100 stitches on the needle, and the pattern says the gauge is 4 stitches per inch, the piece will be 25 inches wide.
If you happen to knit tighter and get 5 stitches per inch, your's will only be 20 inches wide. If you knit loosely, and have 3 stitches per inch, then your's will be 33.3 inches wide.
A big difference if it's a sweater.
To adjust gauge, changing needle size is what's called for.