This is one of those terms wine geeks use to make people's eyes glaze over! Perhaps a culinary analogy will serve to clarify complexity. Imagine you are making a fresh salsa. You begin with chopped tomatoes in a bowl. Taste it. There is only one flavor there: The chopped ripe tomato. Now add chopped raw onion and taste it. Two flavors! Now add diced fresh jalapeno: Three flavors and some spice heat as well. Now add some lime juice, then chopped fresh cilantro (five flavors), and finally sweet bell peppers, giving the salsa six flavors. Somewhere along the way, some of these ingredients will interact with each other, creating entirely new flavors. That is complexity.
Some wines are a one-note melody, some a three-piece jazz combo, some can be a symphony. All have their place, but tickets to the symphony will usually cost more! Wines that exhibit greater levels of complexity are usually much better quality, show higher flavor intensities and demonstrate longer finishes.