What gauge is
Gauge is how many stitches and rows your knitting makes over a set distance, usually 4" (10 cm). A pattern is a set of stitch counts that assume one gauge; knit it at 18 stitches to 4" instead of 20 and every piece comes out a tenth larger — a 40" sweater becomes a 44" one. Yarn weight gets you to the neighborhood; your hands, the needle and the yarn together decide the number, and the only way to know it is to knit and measure.
When you can skip it
A scarf, a dishcloth, a blanket whose size does not matter, a hat you will try on as you go. Anything that has to fit — a sweater, socks, mittens, a hat for someone else — or that must come out of a fixed amount of yarn, does not.
Knitting the swatch
Cast on enough stitches for about 6" of width at the pattern's gauge, plus a few — for a 20-stitch gauge that is 30 to 34 stitches. Knit at least 6" in the stitch pattern the gauge is given in (usually stockinette, sometimes the pattern's cable or lace), with the needle size the pattern suggests, and bind off loosely. Four inches is not enough: the edges of a swatch are distorted, and you need a clean middle four inches to measure.
Knit it the way you will knit the garment — same mood, same hands, not tighter because it is "just a swatch". If the project is in the round, swatch in the round, because most knitters purl at a different tension from their knit and a flat stockinette swatch will lie to you about a tube.
Wash it
Then wash and dry the swatch exactly as the finished piece will be washed, and let it dry flat. This is the step people skip, and it is where superwash merino grows a stitch an inch, cotton relaxes, and a lofty wool blooms and tightens. The pattern's gauge is the gauge after washing; so is yours.
Measure it honestly
Lay the dry swatch flat without stretching it. Put a ruler across the middle, away from the edges, and count the stitches in 4" — including half stitches. Count rows the same way in the other direction. Measure in two places and take the average. A tape pulled tight, or a swatch tugged "a little", is the usual way to get a number that matches the pattern and a sweater that does not.
Fixing it
Too many stitches in 4" (your fabric is tighter): go up a needle size and swatch again. Too few (looser): go down one. A whole stitch per 4" is usually one needle size; the needle size chart has the steps. Change the needle, not your hands — you cannot knit "looser" for a whole sweater.
Stitches right, rows off. Common, and usually fine: most patterns are written in inches of length, not rows, so you knit to the measurement. It matters for a yoke or raglan shaped by row counts; there, choose the needle that gets the stitch gauge and recalculate the rows.
Right gauge, wrong fabric. If the only needle that hits gauge makes a fabric you dislike — a board, or a net — the yarn is wrong for the pattern. The substitution guide is for finding one that hits gauge at a fabric you like.
What else the swatch tells you
Whether the color pools, whether the yarn pills when you rub the swatch, whether it is soft enough against your wrist, whether it grew in the wash and by how much, and how much yarn an inch takes — weigh the swatch and the enough-yarn guide turns that into a yardage estimate for the whole piece. Keep the swatch with the leftover yarn and the ball band; it is the record of the project.