I like the other answers, but I like the question so I wanted to give a more complete and thorough answer. I'd have to say, though I much prefer religious folk contorting their holy books to match science rather than trying to contort science to match their holy books.
So firstly, as Oddthinking points out in his answer, the age of the universe is not some 15 3/4 billion years old. Rather about (13.798±0.037)×10^9 years (as calculated with data measured in 2013 by Planck (space observatory). As I understand it there is a high degree of concordance on this and this number is not likely to be off by much. Now this would actually be impressive if it wasn't so obviously post hoc. Which makes it kind of funny that's it's as far off as it is, even with such a contrived explanation, they couldn't have gotten closer?
But my spine needs some chilling so let's take a closer look...
Day 1 (note this is RSV)
3 And God said, “Let there be light”; and there was light. 4 And God saw that the light was good; and God separated the light from the darkness. 5 God called the light Day, and the darkness he called Night. And there was evening and there was morning, one day.
The first obvious problem is there is the earth. So actually light did exist before the stars. It seems the first photons formed after about 1 micro second. God's a fast talker. But that pesky earth... But there is what's called the photon epoch which lasts from about 10 seconds after the big bang until about 380,000 years. Matter like hydrogen, helium and some lithium was created in this time but no stars yet.
Day 4 (I know 2 comes next, God doesn't)
14 And God said, “Let there be lights in the firmament of the heavens to separate the day from the night; and let them be for signs and for seasons and for days and years, 15 and let them be lights in the firmament of the heavens to give light upon the earth.” And it was so. 16 And God made the two great lights, the greater light to rule the day, and the lesser light to rule the night; he made the stars also. 17 And God set them in the firmament of the heavens to give light upon the earth, 18 to rule over the day and over the night, and to separate the light from the darkness. And God saw that it was good. 19 And there was evening and there was morning, a fourth day.
So on the fourth day God made two great lights (a single star and a.. moon?) and a lesser light (~10^29 "greater light"s). We are firmly still in the first 8 billion years here. Note that stars are still being born and dying but the first stars will be in the first billion years. Star formation is actually quite beautiful, even if there are some artificial colors in there. Note that some elements like gold, can only be produced in the death of a star. So stars were born and died before our planet ever formed. So there is that pesky earth...
Day ? (I have no idea what day to call this, it's all over the place.)
So the next thing that happens is the formation of our star(day 4). This would have happened about 9 billions years after the big bang or about 4.6 billion years ago. A protoplanetary disk would have formed around the new sun which much of would collapse into planets (planets like Venus would probably be day 4, considered stars). The earth would have formed from this (day 1). There is various hypotheses about the origin of the moon but I believe the most accepted hypothesis right now states that a planet, Theia smashed into the proto-earth. The collision ejected the earth's crust into orbit, some of which consolidated into the moon(day 4).
6 And God said, “Let there be a firmament in the midst of the waters, and let it separate the waters from the waters.” 7 And God made the firmament and separated the waters which were under the firmament from the waters which were above the firmament. And it was so. 8 And God called the firmament Heaven. And there was evening and there was morning, a second day.
There would have been some Hydrogen and Helium in the earth from the start, this doesn't tend to stick around. But, as the earth cooled our atmosphere would have formed (day 2) then clouds would have formed and rained down on the earth forming the oceans(day 1). This seems to have been about 4.4 billion years ago(pdf).
Day 5 Sorry day 3, you have to wait until later.
20 And God said, “Let the waters bring forth swarms of living creatures, and let birds fly above the earth across the firmament of the heavens.” 21 So God created the great sea monsters and every living creature that moves, with which the waters swarm, according to their kinds, and every winged bird according to its kind. And God saw that it was good. 22 And God blessed them, saying, “Be fruitful and multiply and fill the waters in the seas, and let birds multiply on the earth.” 23 And there was evening and there was morning, a fifth day.
Ok the Bible isn't interested in the origin of life, that stuff's boring. So skipping forward a few billion years of microscopic life and ignoring the birds for now, we'll come back to them, we find those watery swarms. Animal life seems to have arose around 575 million years ago(pdf) and indeed it was in the ocean. But things really got kickin' about 30 million years later in the Cambrian radiation (a period of about 20 million years where rapid diversification of animal life took place).
9 And God said, “Let the waters under the heavens be gathered together into one place, and let the dry land appear.” And it was so. 10 God called the dry land Earth, and the waters that were gathered together he called Seas. And God saw that it was good. 11 And God said, “Let the earth put forth vegetation, plants yielding seed, and fruit trees bearing fruit in which is their seed, each according to its kind, upon the earth.” And it was so. 12 The earth brought forth vegetation, plants yielding seed according to their own kinds, and trees bearing fruit in which is their seed, each according to its kind. And God saw that it was good. 13 And there was evening and there was morning, a third day.
Skipping over the dry land bit as it was covered in the formation of oceans. We're interested in the plants. But we need to ignore the talk of seeds, fruit and trees. Following a series of extinction events we are brought to the Ordovician era where we see the first plants. Plants will continue diversifying and evolving until flowers emerge after even mammals.
24 And God said, “Let the earth bring forth living creatures according to their kinds: cattle and creeping things and beasts of the earth according to their kinds.” And it was so. 25 And God made the beasts of the earth according to their kinds and the cattle according to their kinds, and everything that creeps upon the ground according to its kind. And God saw that it was good.
26 Then God said, “Let us make man in our image, after our likeness; and let them have dominion over the fish of the sea, and over the birds of the air, and over the cattle, and over all the earth, and over every creeping thing that creeps upon the earth.” 27 So God created man in his own image, in the image of God he created him; male and female he created them. 28 And God blessed them, and God said to them, “Be fruitful and multiply, and fill the earth and subdue it; and have dominion over the fish of the sea and over the birds of the air and over every living thing that moves upon the earth.” 29 And God said, “Behold, I have given you every plant yielding seed which is upon the face of all the earth, and every tree with seed in its fruit; you shall have them for food. 30 And to every beast of the earth, and to every bird of the air, and to everything that creeps on the earth, everything that has the breath of life, I have given every green plant for food.” And it was so. 31 And God saw everything that he had made, and behold, it was very good. And there was evening and there was morning, a sixth day.
[note that the following is a brief simplification and overview of life since the Cambrian radiation to give you a feel for it. It's not as well cited just linked up to Wikipedia. It's uncontroversial so I hope there's no problem].
Some more extinction events bring us to the Silurian. In the Silurian we get diversification of jawed and bony fish and the first terrestrial animals, some arthropods. And then the Devonian where we get land plants and further diversified fish. Another extinction event, then the Carboniferous brings us amphibians. Then after an extinction event we get the Permian where we have the mammals and reptiles, this is capped off with the earth's largest extinction event finally killing off the trilobites. Then we find ourselves in the Triassic where we see the first seeds, corals, and a lot of different (what we might recognize as) dinosaurs. Another extinction event and we are in the well known Jurassic where you get the (true) dinosaurs. In a lovely turn of events no major extinction event happens and we are brought into the Cretaceous, finally it starts looking a bit more like home with the first flowering plants, recognizable insects like butterflies. Then we have probably the most well known extinction event and it brings us to the Paleogene where mammals and birds diversify and we see the first grass. Again, not characterized by a major extinction event we find our selves in the Neogene, birds and mammals continue to diversify and we see the first hominids. This is followed by the Quaternary period which takes us through the last 2.6 million years of the earth's history and of course the development of humans and the domestication of cattle.