A relational database management system is a piece of software which takes organized information and processes it in order to answer questions presented by other software.
In a typical relational database management system, data is arranged into tables, each of which is similar to a worksheet in an Excel workbook. Information in these tables is matched, and processed, and returned to the user. A skilled database architect can arrange the data so that the structure (called the schema) is easily maintained and extended, so new types of information can be stored as the need arises.
Object-relational database management systems take this system and expand it, allowing for more complex data to be stored in each column and for a wider range of calculations to be attached to the table. The combination of more complex data and complex calculations allows one to build advanced databases that can do more than one could before.
These systems are called object-relational because they are an attempt expand what relational database management systems can do by borrowing some ideas from object-oriented programming. The two disciplines are however very far apart because object-oriented programming aims to model behavior encapsulated behind interfaces while object-relational database design seeks to extend relational math to model derived information as well as what is minimally known.