Dandelions are a familiar lawn weed often enjoyed by children but disliked by adults, especially when its bright yellow flowers go to seed. It grows best in sunny, thin lawns, and can tolerate a rather dense, hard soil but not a lot of foot traffic.
Dandelion seeds can blow in from elsewhere and remain viable for years in the soil, waiting for a chance to grow. And because they're perennial weeds, they will keep coming back, larger and stronger each year unless they're removed or killed.
The most effective herbicide for killing dandelions is Triplet which is a combination of 2,4-D; Mecoprop-p; and Dicamba. The single best time to apply these products is in September when the plants are transporting carbohydrates from the leaves to the roots for winter storage. The herbicide is moved internally to all parts of the plant, disrupting its growth pattern. You may not see much response in the fall immediately after application, but it's unlikely the dandelions will sprout again the following spring.