BAS

If you are a business registered for GST you will be required to lodge a Business Activity Statement. This statement will include PAYG for staff and PAYG instalments along with other taxes like Fringe Benefits Tax, Fuel Credits or Wine equalization Tax.

See https://www.ato.gov.au/business/business-activity-statements-(bas)/ for more details. 

PAYG

Pay as you go relates to :

Employee wages

PAYG Instalments

PAYG

You do not have to be registered for GST but employee staff.

As such, a Business Activity Statement will not be applicable. You will need to still declare and submit Single Touch Payroll (STP) for your employees.

Link on the ATO for more details https://www.ato.gov.au/Business/PAYG-withholding/

PAYG INSTALMENTS

If your company/business made a profit in any financial year, the Australian Taxation Office will require you the following year's tax liability in advance. 

Link on the ATO for more details https://www.ato.gov.au/General/PAYG-instalments/

@Andrew Oldham and @Kirsty Fox  curious to read what you would add.