A database is a tool for storing and accessing tables of information. Traditional databases store information in fields and records (columns and rows or attributes and values). The types of data that fields can hold varies across different types of databases but, generally speaking, they hold numeric and text data. The main feature of a database is that of querying, where you can retrieve information that meets your specific criteria. Relational databases allow you to join information from multiple tables using a common piece of information that is in both tables.
To learn more about relational database management systems (RDBMS), see http://en.wikipedia.org/wiki/RDBMS.
A spatial database is much the same, but it can also store geographic data. Several databases and GIS products use the term spatial database to mean slightly different things. For example, ESRI's Spatial Database Engine (SDE) isn't a spatial database, but is advertised as an interface between client software and a normal database. It allows spatial data to be stored in SDE's format, in the database. To load and manipulate the spatial data, you need to have an ESRI product or access to an ESRI service.
ESRI's SDE product isn't the only option for spatially enabling a database. Despite the limited amount of marketing, there are other databases available that better fit the description of spatial database. Oracle has an extension called Oracle Spatial and IBM has a Spatial Extender for ...