I assume you're asking about artificial reproductive cloning, since people don't usually get upset over cloning on smaller levels (cellular or molecular). Reproductive cloning is done by taking an ovum (egg cell) from the surrogate (who will be implanted with the clone) and inserting the nucleus of a cell from the original animal to be cloned. This is given a current pulse to fuse them into one cell and implanted back into the womb of the surrogate.
People get upset about cloning, especially when the discussion is human-related, because there are ethical issues to creating an exact duplicate of someone. While some people argue the theological/philosophical side of the implications of an "artificially created soul", scientists also point out that clones tend to have shorter life spans, which means that if you created a duplicate human, he/she wouldn't live as long as he/she should, and would be prone to more health problems.
There are also some people who argue that you're wasting the potential life in the ovum from which you are replacing the nucleus.