I think that the "conservative" in "conservative republican" refers to economic policies. Someone who supports conservative economic theories believes that the flow of money between the individual and the state should be kept as low as possible ("Read my lips."). This is in accordance with the idea of a free market. A conservative government will tend to take less money from its people and in turn make less provisions for its people. They believe there should be less welfare, etcetera. Contrast this with "liberal" economic theories, which are closer to socialistic ideals. A liberal government will want to have more control of the money in the state it is governing, and will tend to redistribute the wealth among the people of the state. Tax systems under liberal governments try to drain wealth from the wealthy and give it to the poor (in a nutshell).

As for abortion, the environment, gun laws, etcetera there is, I think, little natural disposition towards agreeing with ALL the opinions of some hypothetical average republican (or democrat). However, long before we make up our own minds on such matters we are lectured on what these two "parties" believe. The media would have us believe that there are two mainstream trains of thought. We are led to choose one, and defend it with passion.

Be brave. Break the mold.