Basically there are two known techniques for cloning. The embryo separation technique is actually one that happens in nature. Identical twins are a result of the natural separation of an embryo into two identical embryos. This process can be done artificially by separating an embryo when it reaches the four-cell stage. The other technique that is currently raising the fuss is called nuclear transfer.
Nuclear transfer is a pretty simple concept.
You need two creatures, one to provide an infertile ovum and another to provide a cell nucleus. The one providing the cell nucleus is the one being cloned. The nucleus is removed from the cell and implanted in the infertile ovum. The ovum is then given an electrical shock to fuse the new nucleus into the ovum. The implications of this simple technique are astounding to the imagination, especially when it comes to the possibilities of cloning humans.