Short answer: About 8,000 animal genera [including now-extinct animals]
Bible Context: Genesis 6:19-7:3
“And of every living thing of all flesh, two of every sort shalt thou bring into the ark, to keep them alive with thee; they shall be male and female. Of fowls after their kind, and of cattle after their kind, of every creeping thing of the earth after his kind, two of every sort shall come unto thee, to keep them alive. And take thou unto thee of all food that is eaten, and thou shalt gather it to thee; and it shall be for food for thee, and for them. Thus did Noah; according to all that God commanded him, so did he. And the LORD said unto Noah, Come thou and all thy house into the ark; for thee have I seen righteous before me in this generation. Of every clean beast thou shalt take to thee by sevens, the male and his female: and of beasts that are not clean by two, the male and his female. Of fowls also of the air by sevens, the male and the female; to keep seed alive upon the face of all the earth.”
In Darwin’s day, many Christians had compromised the Bible to accommodate long geological ages. They’d taken the unScriptural position that Noah’s Flood was local rather than global, and that God had created all of the animals as they exist today in their present ecosystems. Some Christians still suppose that God created the plants and animals we see today in their present form. This idea has led them to question the feasibility of Adam naming the animals within a day, of how all of the animals could have fit on the Ark and how a loving God could create Nature, “red in tooth and claw.” In doing so, they fail to account for the effects of the Fall, the world-wide Flood itself and the implications of the Biblical term “kind.”
The Bible states that God created plants and animals “after their kind” and told them to be fruitful and multiply “after their kind.” [Gen 1:12,25] This implies variation within fixed limits, which Mendel confirmed in his studies on heredity and which we observe in nature. A dog is still a dog, be it a dire wolf, fox, English bulldog or chihuahua. We can say then that canines are a “created kind.” Baraminology is an exciting new field which studies such “created kinds,” or baramins. God created the original wolf/dog baramin and gave it the potential to adapt to various environments and conditions.
Note that a created kind is NOT the same thing as a species.
Thus, Adam did not have to name all of the different varieties of canines, like wolves, hyenae, foxes, etc; Adam simply had to name the original canine baramin. The same applies to other animals: He only had to name the original feline, ceratopsian dinosaur, horse, chicken and so on.
At the Fall, the earth was cursed, leading to thorns, disease, death and suffering. Originally vegetarian [Gen 1:30], some animals became predatory or venomous sometime afterward, before or after the Flood. This potential was coded into their genes by the Creator. Natural selection, as a conservative force, weeds out harmful adaptations and uses this God-given genetic potential to allow animals to adapt and survive in a Fallen world. We see evidence in the fossil record of thorns, cancer, death and struggle. If the Bible is true, this means the fossil record would have to have been formed after the Fall. Logically, it follows that the Flood was the cause of the fossil record.
At the Flood, God sent 2 of each unclean created kind [and 7 of each clean kind] to the Ark. This only included air-breathing land animals, fowl, and creeping things. Sharks and dolphins would not have been aboard the Ark.
But what about dinosaurs? If the fossil record, which includes dinosaur fossils, was created by the Flood and God commanded Noah to bring two of each kind of air-breathing land animal, fowl and creeping thing, how many dinosaurs would be required? Fortunately, there were only about 50 kinds of dinosaurs. For example, varieties of the ceratopsian dinosaur kind have different arrays of neck frills, horns and knobs, but Noah would have needed only 2 individuals from this baramin. Likewise, Noah wouldn’t have needed to bring horses, donkeys, zebras and now-extinct three-toed horse varieties, but only 2 members of the equine kind. By this model, John Woodmorappe has calculated that there were probably about 8,000 animal genera [including now-extinct animals] aboard the Ark.
Upon leaving the Ark, the animals again multiplied after their kind [Gen 8:16-19] into all of the varieties we see today, while others went extinct.