I know of a number of approaches to evolution vs creation in Jewish thought. As far as I can tell, it seems that an insistence on the Torah giving literal history with it being roughly 5,769 years since ex nihilo, became more popular after the scientific challenges of the past two centuries, not less. As though we dug in our heels in the face of so many rejecting the Torah for a blind acceptance of the zeitgeist and the importance it gives scientific research.
Here is a list of the approaches I've heard:
1- Rejection of scientific conclusions. Theories change over time. Rather than worry about a contradiction between current theory and the Torah, one can simply wait without concern as science slowly converges to the Torah’s truth.
After all, in the last century theory has gone from Aristotle’s eternal universe to acknowledging that it has a beginning. Compared to that, current difference are small.
This is the basic approach taken by R’ Avigdor Miller.
2- History as a backdrop. As one opinion in the Gemara has it, Adam was created as a fully mature man of 20, and trees were created fully grown, etc… The Lubavitcher Rebbe zt”l concludes that this opinion would hold that the universe as a whole was formed with a history consistent with a natural, scientific, progression.
One may then ask why Hashem chose to create a world that has an artificial age. Or perhaps not: Can one understand why G-d chooses to do anything?
Personally, I have adifferent problem with this position. How does one ascribe a time to creation? It can’t be on the Creator’s clock, since He Exists outside of time. Therefor, when we speak of “when” creation happened, we mean the beginning of the universe’s timeline. So then how could we talk about G-d creating the universe at some point in the middle of the line, allowing history to go in both directions — past and future — from that point?
3- Conflict resolution. Invoking relativity or whatnot to show that 15 billion years can be 5758 years in another frame of reference. Perhaps relativity justifies the differences between frames of reference (as suggested by Rabbi Yaakov “Gerald” Shroeder). The “birds” of day 5 are actually dinosaurs, which are most similar biologically to birds of any thing living today. Creation of the sun on day 4 is actually about the sky clearing to the point the sun could be seen on earth, etc…
As can be seen from my treatment, I don’t consider this opinion fair to either the Torah or the scientific data. Yet, many popular books have come out in the past two years promoting this kind of position. Perhaps someone else can do it justice.
[The next two paragraphs are minor paraphrases of material R’ Gil Student wrote for his Hirhurim blog.]
On the other hand, however Bereishis 1 is understood, there is a poetry to the idea. In Collected Writings VII pp 363-264, R’ Samson Raphael Hirsch rejects an insistence on literalism, common to both of the previous approaches, and waxes poetic about the greater display of Divine Wisdom a natural unfolding would suggest. Similarly, Rav Kook makes the same point in Orot haQodesh 91.
R. Menahem Kasher, in Torah Shelemah (Bereshis, ch. no. 738), quotes a responsum from the Geonim in which it is stated that Adam was first created as a speechless creature, like an animal, and only later was given speech. This could certainly be interpreted as a precedent for the claim that Adam was descended from humanoids. R. Kasher suggests that this is a matter of dispute between the Ramban and his student R. Bahya ben Asher, with the Ramban on the side of the Gaon’s responsum. In his Hibah Yeseirah (Bereshis 1:26, printed in the back of Bnei Banim vol. 2), R. Henkin writes explicitly that Adam’s body was taken from creatures that preceded him and it was only his soul that was created ex nihilo. In other words, Adam evolved from lower creatures and became human when God created and implanted in him a human soul.
4- Multiple creation times. This is the approach of the Tif’eres Yisrael. He cites an opinion of the tannaim, a central theme amongst the more kabbalistically inclined rishonim, that Hashem created worlds and destroyed them before this one. Dinosaur bones and starlight are legacies of these earlier worlds. The Tif’eres Yisrael did not say anything about evolution, just that this earlier time explains what the fossils are fossils of. In Techeiles Mordechai, R’ Shalom Mordechai Schwadron speaks laudably of the Tif’eres Yisrael’s resolution.
In Gen 1:1, G-d creates ex nihilo (matter from nothing). Then, before verse 2, these other worlds (in this opinion, epochs) rose and fell. Then, there was “chaos and emptiness” from which our world emerged. The universe as a whole, even the planet, can therefor be older than 5758 years.
Since current theory is that the world started as a singularity — IOW, not within the purvey of science, it is all a matter of faith if the ex nihilo was with the intent of the Creator or not.
Rabbi Aryeh Kaplan quotes R’ Yitzchaq meiAkko (a student of the Ramban) who concludes from the Zohar that the first creation was 15.8 billion years ago — the age astronomers and physicists seemed to be converging on in the 1980s and 1990s, given multiple ways of measuring the age. It is unclear that this is truly the intent of R’ Yitzchaq meiAkko, but that’s Rabbi Kaplan’s take. The original lecture to AOJS, which is more complete and persuasive than the mention in his NCSY book, is available on line. This is built on an idea discussed by Rabbeinu Bachya and numerous other kabbalistically inclined rishonim, that of our world being one of a cycle of shemittos, so that there is history and time before our universe.
5- Rejection of a literal read of the Torah. This is much easier, halachically, than it sounds, as there is a long tradition, including the Rambam and the Vilna Gaon, teaching that Genesis 1&2 actually convey deeper truths via metaphor. The gemara, after all, limits the number of students (to 2) that one may teach the secrets of the Act of Creation — so clearly we can’t just take the text at face value.
Another commonly cited proof for non-literalness is that the word “day” precedes the creation of the sun. Therefor, it can’t be used, at least in this narrative, to mean our 24 hour period.
The Rambam takes “day” to mean a stage in the causal chain, not a reference to time altogether. See The Rambam on Time During Creation.)
6-The Maharal (intro to Gevuros Hashem) teaches that creation is so alien to human experience that we don’t have a comparison to it. Therefor prophecy, which is transmitted by visions, can not describe it. (The World to Come is similarly explained. This is why it only appears in Tanach as “your days will be prolonged”. Continued existence we can understand. The rest of the details, no.)
However, creation is also so alien that we can not understand it by extrapolation, either. In general, the Talmud teaches that “wisdom is greater than prophecy”. The Maharal explains that this is because the power of extrapolation and deduction takes you further than just what can be presented metaphorically in visions. In this case, though, creation is beyond wisdom as well — which is why the Talmud limits the forum where it can be studied.
The Maharal's conclusion is that the Torah can’t provide us with a comprehensible history AND that science must be wrong.
R’ Dessler (vol. II) ascribes a similar opinion to the Ramban, at least with regard to time during creation. That the days of creation were both literal days of seconds, minutes and hours, but also the subsequence six millenia. Not that they represent or parallel the subsequent millenia, but they are literally the millenia themselves. These two perspectives appear to contradict, but only because of limitations of how humans perceive time ever since eating from the tree of knowledge. (A longer description of Rav Dessler’s opinion can be found at Rav Dessler’s Approach to Creation.)
In the two last opinions, the presumption must be that Genesis 1 and 2 teach some deeper truths about reality. Either because that’s the only meaning of the text, or because all we can understand from the text are partial truths that don’t quite add up to a whole picture. In either case, without having a metaphor, there would be little reason for its inclusion in the Torah.
The Maharal explains some of the symbolism of the number 7 later in Gevuros Hashem. The seventh should be made holy even without the creation story, so it is possible the details of the story are made to describe this point.
One can also see a pattern: light, sky-and-sea, earth; repeated twice. First Hashem created light. On day four, He created the stars, moon and sun — the sources of light. Second, He separated sky from the sea. On day 5, He created those who live in the sky and the sea — the birds and the fish. Third, Hashem made the seas converge to show land. On day six, the animals and people inhabited the land.
What is important to us as Jews is not what actually happened, that is, whether G-d used natural or miraculous means to create the universe. Rather, to take the lessons of creation, or the lessons encoded into the story of creation, and live them.