A complication is any function a watch performs beyond simply telling the time. The date window on your watch is a complication. So is a stopwatch, a second time zone, or a moonphase display. The word sounds intimidating, but it just means "an extra feature" — and the more of them a watch has, the more "complicated" it's said to be.
Common complications you'll actually see
Most watches you'll shop for have at least one. A date is the most common, sometimes paired with the day of the week as a day-date. A chronograph adds a stopwatch. A GMT tracks a second time zone. A power reserve indicator shows how much wind is left. Fancier watches add things like a moonphase, which tracks the lunar cycle, or a perpetual calendar that knows the length of every month. For everyday wear, the date and maybe a day are all most people use.
More complications isn't always better
Each complication adds parts, cost, and things that can eventually need service. A busy dial can also be harder to read and, in a mechanical watch, more expensive to maintain. A clean watch with just the time and a date is often the more elegant, more legible, and more affordable choice. Complications are wonderful when you'll actually use them — a traveler and a GMT, say — but they're not a measure of how "good" a watch is.
Choosing the right ones for you
Pick complications based on your life, not the spec sheet. Travel a lot? A GMT earns its place. Want the simplest daily watch? A date is plenty, and a no-date dial is even cleaner. The best watch isn't the one with the most functions — it's the one whose functions you'll genuinely use.