How do new species evolve?

Allopatric speciation:

New species appear when one interbreeding population is split into two reproductively isolated populations, which then evolve to become separate species. This mostly happens when populations become geographically isolated, for example by mountain chains forming, so that the two populations are physically separated and can no longer breed with each other. This process is called allopatric speciation.

Evolving into separate species while the populations are isolated can happen because they are adapting to different environments, or simply through chance events that change the frequency of certain genes in each population. Even if the reproductive isolation is not complete when the populations meet again after being isolated, so that they can still interbreed, a process called reinforcement may still finish the process. Reinforcement happens when any hybrid offspring are less able to survive than their parents, so that the populations stay reproductively isolated due to natural selection, and evolve into completely separate species.

Sympatric speciation:

Speciation can potentially occur under certain circumstances without geographic isolation, which is called sympatric speciation. An example of this happening is the cichlid fish in Lake Malawi, which have diversified into hundreds of different species from a single ancestor. However sympatric speciation is strongly debated, and how important this process might be remains unclear.

Hybrid speciation:

Plants can often form new species through a process of polyploidy and hybridisation. Polyploid cells or organisms have more than the two paired sets of chromosomes you receive from each parent. This mutation often makes the new plants unable to have fertile offspring with normal, diploid plants with only two chromosome sets, creating a new species. Hybrid speciation happens when the hybrid offspring of two separate species form a new species, which is reproductively isolated from the either parent species.