DISTILLATION
Condensation of evaporated water is the simplest method of desalination - and is if fact the primary method used in nature, where ocean water is evaporated, carried to cooler climates where it condenses and falls out of the sky as rain.
Distillation is simply the man-made version of this natural process. It involves heating water to its boiling point, evaporating a portion of this water, and condensing the resultant steam. However, in desalination, only a portion of the water is boiled and condensed - with the balance of water containing the salts at a higher concentration - is discharged as a waste brine stream.
Since boiling water requires a large amount of thermal energy, practical systems use a number of heat exchangers connected in a counter-current fashion to recover the heat from the condensed steam - thereby substantially reducing the thermal energy required.
Several variations exist. Multi-Stage Flash (MSF) operates at just below the normal boiling point of water, with boiling in a number of counter-current stages operating at just slightly below atmospheric pressure. Multi Effect Distillation (MED) operates at a lower temperature, with a lower pressure requirement, VTEMED which is merely a MED design using verticle tubes, and Vapor Compression operates at the lowest temperature - with all of the energy input being electricity required to drive the compression pumps. There are also variations with respect to the heat transfer equipment, some using tubing versus flat plates, verticle vs horizontal tubing, submerged tubing, etc.
While the heat to drive such plants may be supplied by any combustion system, most existing large scale systems derive their heat from the condensation of steam in the final stages of a coal, gas or nuclear power plant. This is particularly advantageous in that not only is the large amount of thermal heat essentially free but also because the top heat exchanger in the desalination facility replaces and negates the need for a similar condenser in the power plant to condense the power plant steam for recycle back into the power plant.
Small scale facilities driven by solar panels have recently become commercially available and conceptual designs exist for larger scale facilities.
It is worthy of note that the amount of thermal energy required is essentially independent of the initial salt concentration (since the salts do not either add much to the energy required to bring the water to the boiling point or significantly increase the energy of vaporization at the boiling point.).
As with all systems handling water with contaminants, special filtering and sometimes the use of additives is done to minimize corrosion and solids build-up on the many heat exchange surfaces.
STANDARD MSF PLANT DESIGN
The energy to raise sea water to the boiling point, and the additional heat of vaporization, is quite large and would be very expensive.
To surmount this problem practical MSF plants use multiple heat exchangers connected counter-currently to recover the heat from the condensing steam and use it to heat up the incoming feed water. Thus, the only additional heat needed is added at the top of the sequence, and is equal to only the heat losses in the system. Thus, the primary trade-off is the savings in heat input versus the added capital and operating costs of this substantial amount of heat exchange surface area.
An additional typical solution is to utilize what would otherwise be waste heat from an electrical generation plant - by using the heat of condensation from the power plant exhaust steam to pre-heat the incoming feed water.
The following is a sketch of a flow diagram for a typical MSF desalination plant using waste heat from a conventional steam based electrical power plant.

There are many of these plants, including many large scale plants, in operation in Saudi Arabia and related areas where water is scarce and where waste heat from electricity generating plants is readily available.