Gandhi, in one of his more famous writings, characterized seven "Sins in the Modern World". One of them was Science without Humanity. Stem cell research holds great promise for being able to treat or prevent any number of diseases and disabilities. But altering or "messing with" the fundamental building blocks of the human species is clearly an area we do not fully understand. We do not know what the long term implications are yet, and so we must proceed cautiously, and not get too blinded by the scientific and medical potential of such research.
People who are against stem cell research believe that it is wrong to play God. They will say that it is immoral to mess around with human life. They do not find that it is ethical to conduct research on embryos.
Embryonic stem cell research is what many people disagree with. This was the main form of stem cell research up until recent years. Since then, research has started to move on to a different kind of stem cell research called iPS (induced pluripotent stem cell). Pluripotent cells are artificially derived. An example would be an adult somatic cell. This allows researchers to use stem cells without having to get them from embryos.
The reason that many people disapprove of the idea of stem cell research is that embryonic stem cells come from human embryos. These embryos have the ability to become living human beings.
This means that studying stem cells involves destroying what could have become living humans. To many social conservatives, an embryo is just as alive as a living person and has just as much right to live. So, to them, killing an embryo so as to use its stem cells is no different than Nazis experimenting on living human beings.