Cloning is the process of creating an identical copy of an original. A clone in the biological sense is an organism that is genetically identical to another living organism. There are two main types of cloning: cloning an organism or cloning a gene.
Cloning an organism means to create a new organism with the same genetic information as an existing one. This can be done by a process called somatic cell nuclear transfer (SCNT). The nucleus is removed from an egg cell and replaced with a nucleus extracted from a cell of an organism one wants to clone. As the nucleus contains (almost) all of the genetic information of a lifeform, the host egg cell will develop into an organism genetically identical to the nucleus donor. This is how Dolly the sheep, the first mammal ever cloned, was made: from a single cell taken from the udder of her mother.
Cloning a gene means to extract it from one organism and to insert it into a second organism where it can be used and studied. Cloning a gene sometimes can refer to genetic engineering in a gene associated with some phenotype (physical feature or trait) that can be transferred from one organism to another.