Your SSMS has nothing to do with it. SSMS is just a tool with which you connect to an SQL Server instance. That SQL Server instance could be running on your local machine, or it could be running on a different machine. You may have permissions to create databases in one instance, but not on the other.
If you are only playing with SQL Server at home, you are likely to have SQL Server on your local machine. In that case, you are likely to be sysadmin on the instance, but it depends on how you installed SQL Server. When you install SQL Server, you have to specify a login that is to be sysadmin on your server.
If you are in a corporate environment, and you are connecting to a SQL Server instance in that network, you will indeed have to speak nicely with the DBA to get permissions to create the database. (For which you do need need to be sysadmin, but you need the permission CREATE DATABASE or membership in the dbcreator role, if memory serves.)