Republicans are against Obamacare because it drives up the cost of health care for businesses, the party sees it as a possible violation of the Constitution and there are tax increases attached to Obamacare. There is also the idea that Republicans are simply afraid that people might like Obamacare.
A business owner who has to pay more for health care because of Obamacare might be tempted to hire more part-time employees so he doesn't have to offer those employees health benefits. One argument against this is that businesses have been paying more for years to offer employees health benefits while putting more of the cost of those benefits on employees.
According to the U.S. Supreme Court in 2013, Republicans are wrong in asserting that Obamacare is a violation of the Constitution. There are also some Republicans who are automatically against anything that President Obama is in favor of.
Research has shown that people actually like the concepts that Obamacare represents, but not the name Obamacare itself because of the association with President Obama. Certain members of the Republican party feel that less government is better, but the concepts contained in Obamacare might make people feel that more government is better, which might cost the Republican party some of its members.