Depends what you are trying to do... What programming language and server environment are you going to be using it on? What coding frameworks (if any) are you using?
PostgreSQL is scalable and can handle terabytes of data, it supports JSON, it contains variety of predefined functions and number of interfaces. The disadvantages are like configuration is confusing, documentation can be spotty, speed may vary during large operations. MySQL offers a lot of functionality, contains variety of user interfaces, it can work with DB2 and Oracle. Disadvantages are like no built-in support for XML or OLAP, needs lot of time and effort to get MySQL.
Almost every hosting provider gives you Mysql DB access with your hosting account, so if you want to develop something that everyone can use go for Mysql. PostgreSQL is more advanced and complex it can be a replacement for high end DB systems like Oracle or MS SQL, but probably you will have to install it yourself on VPS or full server account. Unless you have a very specific need you can safely go with Mysql or it's clones like MariaDB. Good luck.