Is there any database in QGIS? I use ArcMap with a file geodatabase. But now, I have to start a project in QGIS and I don't know, how to create a database.
You can use a number of data formats within qgis, including PostGIS, based on PostgreSQL. Here's a topic describing how to get started with PostGIS.
Have you worked through the QGIS training modules? I am not terribly familiar with QGIS either but it seems that QGIS relies on relational database management systems to create databases and access spatial data. I am currently training myself how to use QGIS here are the links to the training about RDBMS for QGIS.