It is now compulsory to wear a mask in shops and other enclosed public spaces in Northern Ireland.
The decision was taken by ministers last week as part of efforts to suppress coronavirus. Infections in NI have risen three-fold since early July.
Stormont said it wanted enforcement to be light-touch but warned fines of up to £60 were possible.
The Department of Health said it would be an offence to breach the law without having a "reasonable excuse".
Breaches could attract a fixed penalty notice of £60 (reduced to £30 if paid within 14 days), it added.
The amount increases if there are further offences.
The department said that while there is no legal requirement to wear a face covering in a church, "congregations may of course do so voluntarily, and this would always be advisable to protect the health of fellow worshippers".
First Minister Arlene Foster said the executive would have "failed" in explaining the importance of people wearing face coverings, if the situation arises where fixed penalty tickets are being issued.
Face coverings also became mandatory in shops and other indoor settings in the Republic of Ireland from Monday.
Those who do not comply face fines of up to €2,500 (£2,250) or six months in jail, although the government has said police intervention would be a last resort.
Questions remain about how Northern Ireland's new legislation will be enforced.
The PSNI has said the primary responsibility for enforcing the wearing of face coverings will lie with shop owners.
The force said it would "only use enforcement as a last resort, when all other approaches have been unsuccessful".
The chair of the Police Federation for Northern Ireland said it was "not practical whatsoever" to expect police to enforce every breach or potential breaches.
"They'd be running from pillar to post trying to do this," added Mark Lindsay.
"Unless you have a police officer standing at the door into every retail premises, you're not going to be able to enforce it so there has to be an element of common sense."
Retail NI said it would be the "individual responsibility of shoppers to comply" with the new law.
"We hope that shoppers will show understanding and respect to each other and remind them that a significant number of people with a medical condition are exempt from wearing face covers," said its chief executive Glyn Roberts.
"From the start of this crisis we have always said that our main ask of shoppers has been to think of others and be kind," said Aodhán Connolly, director of the Northern Ireland Retail Consortium.
"That has never been more important than it is now.
"We have seen abuse and violence against staff rise and we would ask for the public to remember that our front-line colleagues are doing a great job in tough circumstances."
Roger Pollen, from the Federation of Small Businesses, said enforcement would not involve a "big stick and being heavy handed".
"It is about quickly encouraging people to adopt this as the new norm," he said. "Our members are behind this, supporting it and working with it."
The executive has launched a public information campaign - Wear one for Everyone - that will be rolled out over the coming weeks to raise awareness of the benefits of face coverings.
DUP MP Sammy Wilson has criticised the move, and said it would "create more obstacles for hard-pressed retailers and sap confidence from many who do not feel comfortable in a mask".
The executive said face coverings were a "vital defence in our battle against Covid-19".
"Now that the lockdown has been eased, we must keep following public health advice, whether that be wearing face coverings, maintaining social distancing or washing our hands," it said.
"We need to maintain the collective spirit that got us through the first surge of the virus."
The statement added that face coverings should increasingly "become the norm" but added that some people cannot wear them and there should be "no stigma attached to anyone in these circumstances".
The executive said businesses would be expected to encourage and promote compliance.
It has been compulsory to wear a face covering on public transport in Northern Ireland since 10 July.
On Monday afternoon, the PSNI said no fines had been issued, so far, for breaches on public transport.
Seventy-six new cases of coronavirus were diagnosed in NI over the weekend, according to the Department of Health's latest update.
One more death has been recorded, which happened on 8 August.
The figures, which mostly record fatalities in a hospital setting, show the number of deaths now stands at 557. The total number of confirmed cases of Covid-19 is now 6,140.
Health Minister Robin Swann said news of the death was "very distressing".
"I urge everyone across Northern Ireland to do everything they can to keep themselves and others safe. We all have to remain on high alert," he said.
"We have a duty to protect those in our community who are most vulnerable to this virus."
In the Republic of Ireland, a further 57 cases were announced on Monday, but no more deaths linked to the virus.
What about schools?
The first groups of students in Northern Ireland are due to return to school in the next few weeks - but it is not clear whether they will have to wear face coverings.
Last week, Education Minister Peter Weir said the executive was not recommending face coverings for pupils - but NI's Chief Scientific Adviser Prof Ian Young later said he believed it would be of benefit for older children.
In the Republic of Ireland, ministers have recommended that secondary school students and teachers should wear face coverings when a distance of two metres cannot be maintained.
Do face coverings work?
World Health Organization (WHO) advice says non-medical face coverings should be worn in public where social distancing is not possible.
Coronavirus is spread when droplets are sprayed into the air when infected people talk, cough or sneeze. Those droplets can then fall on surfaces.
The WHO says there is also emerging evidence of airborne transmission of the virus, with tiny particles hanging in aerosol form in the air.
Homemade cloth face-coverings can help reduce the spread from people who are contagious but have no symptoms or are yet to develop symptoms.
Taking a face covering on and off can also risk contamination, the WHO says.
This article was changed on 17 August 2020 to include guidance from the Department of Health the following Department of Health relating to churches and face coverings.