Simple organisms are relatively easy to clone. In some cases entire cells can be inserted into bacteria or a yeast culture that reproduce asexually. As these cultures multiply, so do the inserted cells.
Animals cells are comparatively harder to clone. In the past, animal cells have been cloned through several methods. One of these methods includes inserting the nucleus of an animal cell into a fertilized cell with its own nucleus removed. Another method is to divide embryo tissues and insert them into surrogate mothers where they then develop normally.
Cloning cells requires extremely precise and minute technology. When scientists from ACT (Advanced Cell Technologies) created the first human embryo, they removed the DNA from the nucleus with a needle less than 2/10,000th of an inch wide. Up until very recently, this type of technology wasn't available to us. However, science is expanding at an amazing rate and allowing us to dive deeper and deeper into the composition of cells and even make changes to them.
The diagram below (from millerandlevine.com) shows exactly how Dolly the sheep was cloned. This was a very complicated, multi-part process, and only the major parts of the process are shown here.