I ask this question in all earnestness and with a dash of facetiousness because of the confusion that may be caused by the differing usages of conserve, conservation, and conservative. Especially in our current, post "Earth Day," era; where these words have taken on expanded meanings, beyond the one previously used in the political realm only. Since that first "Earth Day," the word conserve has meant "to use less of," especially in reference to energy and natural resources. Most of us have adopted at least some habits because of what we have learned since then, so we could be asking, "I conserve gas, electricity, and water; what the heck does a conservative (want to) conserve?"
In basic terms, Conservatism's job is to defend and maintain the "aristocratic order," the social hierarchy of whomever is on top of the social strata at that moment. So, even though the "Conservative Movement," is more than 200 years old, and was started in response to the French Revolution, the current crop of "conservatives" is defending a different group of "aristocrats," than Edmund Burke was. Instead of the royal families in Europe, the current crop of US Republicans is defending the "aristocratic order," for US based corporations and the people who run them.
The secondary job of those leading the Conservative Movement is to get the "lower orders," the bottom tier to support them in this endeavour by convincing them that maintaining the status quo, or returning to the previous state of affairs is to their benefit. In the case of the current resurgence, this has been done by evoking the idea of the loss of something they never had. A truly equal chance.
Although conservatism as a movement is more than 200 years old, and has not always been in command of the narrative; it has always surged in popularity each time an era passes that expands democracy, and empowers more and more of that "lower order," of the (formerly) powerless. It is always in reaction to this expansion of democracy because of the loss of power that comes when those who used to be your slaves, used to be your trodden upon laborers, who used to be the ones you could deny basic accommodations to, and especially the ones you used to be able to exclude from the electoral process are now no longer slaves, no longer unrepresented laborers, and no longer silenced at the ballot box.
So, after the greatest expansion of democracy this country had yet experienced from 1955 to 1970, a disgruntled group of folks who benefited from that previous way of life set out on a mission to push back against these changes. It "just didn't set well," with them that they now had to 1) share the bus, 2) sit next to people who had previously been relegated to the kitchen or "out back," 3) send their children to schools that were now going to be integrated, 4) have to pay women as much as they had paid men, 5) hire folks they previously had been able to simply deny an application to, 6) rent apartments or sell houses to people based on qualifications that weren't related to race, religion, or marital status, 7) allow anyone with the proper qualifications to enter university, and 8) not deny credit to someone based on gender or marital status. Whatever the case, the current crop of conservatives is fighting to rescind every one of the emancipatory achievements of the last 65 years.
This current group of Republican conservatives has convinced reasonable numbers of previous Democrats that this new equality for women, blacks, and other minorities was limiting their own ability to succeed in life. And the push back began. It started with "All in the Family's" theme song, "Those were the Days," particularly the line, "when girls were girls and men were men." It was codified with the Hyde amendment, denying certain women access to a newly available medical procedure. It spread to unions under Ronald Reagan. It went on to attack the concept of affirmative action. It has continued to this day in its attack on immigrants. Attack after attack, and those previous Democrats still kept voting against their own best interests because they believed that the Conservatives were on their side, and were going to protect them from those who would take away their "fair shot." The attacks on women's health issues from the Republican Congress and the renewed vigor with which Republican governors took an axe to union rights with this past spring, changed voting laws, and enacted other laws denying democracy shows what they had in mind. It is a whole lot easier if you can convince a group of people to voluntarily give up the level playing field, or for them not to try to get one, than to have to go to war over the issue; and that's just what they did. Now we have a whole bunch of previous Democrats (and some Republicans) with "buyer's remorse."
So, what is it the the Conservative Movement is trying to conserve? They are trying to conserve a life for the "Top Tier" that does not include a leveling of the playing field or an opportunity for equality for anyone not in that social circle.