In general: You have to configure the new Redmine instance to use a database. The database may be on the same host or on another.
You can use the old database and forget about the backup (just keep it for the case something goes wrong so that you can recreate it). But you can also create a new database in any (MySQL) DBMS and insert the backup there. I assume the backup is a MySQL dump.
Next, if you have got a database with the old data and if you have a Redmine 3.x instance that is configured to use this database, you have to do the migration: that is executing the Redmine migration. This will modify your old data (create tables/columns, moving data from here to there, ...) so that when the migration has completed Redmine 3.x is ready to use.
Once the database was updated don't use it with an old Redmine instance. I've never tried it, it may not be possible anymore - not sure. In worst case this could corrupt the data.
Database Migration is the key word. Have a look into the doc (fr, de, ru, jp).