Cultural behavior is behavior exhibited by humans (and, some would argue, by other species as well, though to a much lesser degree) that is extrasomatic or extragenetic—in other words, learned.
There is a species of ant that builds nests made of leaves. To build a nest, some of these ants pull the edges of two leaves together and hold them in place, while others carry larva in their jaws and 'sew' them together with the silk they secrete. This is certainly a complex feat of engineering, but it is not cultural. This behavior is instinctive, built into the ants' behavior mechanisms. They cannot alter their plans or think of better ways to join leaves. They cannot teach or be taught to do so.
But there are examples of animals that can learn behaviors, such as dogs and cats. A dog doesn't know instinctively not to urinate or defecate indoors, but it can be taught not to do so. Dogs are capable of learning specific behaviors.
A dog's acquisition of a behavior satisfies one of the requirements of culture, but it also fulfills another. If you were to take a dog that has learned not to eliminate indoors to a different house, it would still know not to urinate there. This is because the dog has made a generalization. It knows not to urinate or defecate in any house, not just the one in which it was taught. However, this behavior only makes two of the four requirements.
For a behavior to be considered cultural it must be shared extragenetically; that is, it must be taught. If a trained dog is introduced to a puppy that doesn't know not to urinate in a house, it cannot teach it not to do so. A particularly intelligent puppy might eventually learn not to eliminate in people's houses by observing the older dog, but no active teaching would have taken place.
Contrast this with an observed group of macaque monkeys. Some scientists wanted to learn about eating behaviors in macaque monkeys, so they put some sweet potatoes on a beach near where they lived. The sweet potatoes got sandy and, as the monkeys disliked dirty food, they would spend some time picking the sand off. One young female, however started taking her potatoes to a freshwater pool to rinse off. She showed the others how to do so as well. The scientists then threw wheat on the sand, hoping the monkeys would spend more time picking the food out so they would have more time to observe them. The same young female just scooped up handfuls of wheat and sand and dumped them in the water. The sand sank and the wheat floated, which she ate. This practice also quickly spread through the group. This is what humans could call a proto-cultural behavior. It is learned, it involves concepts and generalisations, and it is taught. There is only one thing missing.