How is a magnetic field generated?

Magnetic fields have always been generated by moving charges.
For example, a current constitutes moving charges and I hope you might be familiar with the concept of an electromagnet which is a magnetic field being generated around a current carrying conductor on account of moving charges inside it.
When you consider a bar magnet. The tiny electrons moving inside the atoms of the magnet are so arranged that the effects due to each electron is added up to form a noticeable magnetic field.
Another source of magnetic field is an electric field changing with time, i.e. a time varying electric field.