The Government has issued a list which is available on gov.ie which sets out, among others things, guidance/direction as to purposes for essential travel (beyond 5Km), essential retail outlets, essential services, essential wholesale and retail trade and support activities which is available here: https://www.gov.ie/en/publication/c9158-essential-services/.
The measures are made based on expert advice from the National Public Health Emergency Team (NPHET) and are kept under ongoing review by that group, with the over-riding consideration being the protection of public health in the context of the current Covid-19 pandemic. It is a matter for each business to review these and see what applies to them. The current restrictions are scheduled to last until 1st December, at which point the NPHET will issue guidance on whether any changes are considered prudent or necessary.
The current list of essential services does not include dog grooming or dog training services or facilities but does include the provision of veterinary, animal welfare and related services.