A wine is characterized as “complex” when it energizes your palate. A complex wine will keep divulging new flavors, textures or aromas about itself.
Mind you there’s nothing wrong with a predictable, forthright wine that you enjoy. However a complex wine will keep you guessing and will make the wine tasting experience more entertaining.
Complexity is nice, vague (but convincing) term to use to communicate your appreciation for a quality wine. Higher caliber wines tend to be more complex than their more common, mass produced counterparts.
In a nutshell, complexity is just another way to say that the wine stirs your senses.