There are many ways to set up a saltwater tank. Smaller tanks are generally set up skimmerless and sumpless. The bio-filtration is done entirely using live rock and live sand. Mechanical filtration is either omitted or performed by a HOB filter.
In larger tanks, the mechanical filtration is usually entirely by the skimmer. The Berlin Method uses only live rock and live sand for the bio and only a skimmer for the mechanical. This is typically the method used in FOWLR (fish only with live rock) tanks. There is another variation called DSB (deep sand bed) that uses a deep bed of live sand to help consume the nitrates. It seems that it is falling out of favor because macro algaes need a balance of phosphates and nitrates. In the absence of nitrates, the algaes don't effectively consume phosphates.
Lastly, you can buy live rock and live sand so that you can avoid the cycling process. To start up a saltwater tank cold from base rock and bare sand, it is said to take upwards of six months before it is ready for some fish and corals.