a variety of flowers
This refers to a set which contains several kinds of flowers.
Ex. "The bouquet was made up of a variety of red flowers." (They're all red and part of one set, but are different from each other in terms of genus, species, cultivar, and so on.)
varieties of flowers
This refers to multiple sets, each set containing many instances of one kind of flower.
Ex. "Several varieties of flowers are available for purchase." (A flower shop offers several different kinds of flowers, and they sell more than one of each.)
a variety of flower
This refers to a subset within the set of the whole, as in one kind of flower among all kinds of flower.
Ex. "A variety of flower may hold they key to curing cancer." (One particular kind of flower contains compounds that have some effect against cancer, but flowers in general do not have such properties.)
varieties of flower
This refers to several distinct subsets within the set of the whole. Ex. "Several varieties of flowers bloom all year round." (There are several kinds of flower that blossom throughout the year, but there are also those that do not.)