Thanks for the great question!

The ultimate goal of the DNA that makes up our genes is to reproduce: to transmit copies of DNA into the future.

Different living things use different strategies to reproduce. Asexual organisms, such as many species of single cell bacteria, reproduce by forming clones, or exact copies, of themselves which split off from the parent. On the other hand multicellular organisms, such as plants and animals, reproduce sexually -- through the fusion of sex cells. Species evolved to use whichever strategy worked best in their environment.

Also, many living things have evolved to use both asexual and sexual means of reproduction. For example aphids, slime molds, sea anemones, and some starfish are capable of both asexual and sexual reproduction. Typically, when the environment is rich and abundant with resources, these species will undergo asexual reproduction. However, when environments become harsh, and resources are scarce, these species may switch to sexual reproduction. This is because sexual reproduction increases the genetic variability within a species, perhaps allowing them to adapt with their changing environment.

There are far too many to list them all. All animals can reproduce sexually, but some can do both ways. Colonial animals such as siphonophores, corals, and bryozoans are particularly good at asexual reproduction, but like everything else they can reproduce sexually. However, asexual reproduction is known in some lizards, some molluscs, many parasites, and some other animals, too.

In the future, humans might join the club if we start going in for cloning in a big way.