A trade zone is a contiguous set of provinces whose trade posts are owned by the same character. Trade zones give a bonus to city income and trade post income. Trade zones scale super-linearly, such that it's better to have a single 6-province trade zone than two 3-province trade zones.
Patrician family trade zones can be viewed by clicking on the trade zone map mode (G) and then clicking on a republic while holding down the Ctrl key.
Contiguity
For trade zone purposes, control of a sea province is determined by which family owns the most trade posts pointing into that sea province. If there is a tie, the sea zone is considered neutral: no family owns it, and no trade posts can be connected through it. If the sea zone is controlled by a family, all trade posts in adjoining land provinces are connected, regardless of port locations.
Trade value
The value of the trade zone is based on the income and holdings in the provinces with trade posts. Trade zone value also receives a large bonus from being connected to the republic's capital. Some buildings increase trade value as well.
Income bonus
Cities and trade posts within a trade zone receive a percentage bonus to their income depending on the trade value (TV) of the trade zone. Note that this bonus is halved for cities.
Trade Zone Bonus = (TV * 100) / (TV + 200)
Disease spread
With 
Reaper's Due DLC, trade zones can help spread epidemics. Wealthy rulers should consider fortifying their trade zone provinces with upgraded hospitals.
Technology spread
Technology spreads unidirectionally from trade posts to the merchant republic capital.