LATEST: Health Minister Simon Harris has confirmed the country has entered the delay phase of its response to the virus.
That means the infection is being transmitted between members of the community rather than limited to those who have been to Italy or other blighted regions or their close contacts.
The drastic steps aim to reduce the peak impact of the virus and slow its spread, relieving pressure on the health system.
Mr Varadkar said many more people in Ireland would fall ill with the infection.
And he added: "Unfortunately we must face the tragic reality that some people will die.
"The virus is all over the world, it will continue to spread but it can be slowed."
He said the Government has a duty to protect those at-risk categories of citizens, such as older people and those with underlying health conditions.
"We said we would take the right actions at the right time and we have to move now to have the greatest impact," he said.
School closures are among restrictions coming into place at 6pm on Thursday.
People have been urged to continue to go to work but, where possible, do so from home.
Public transport will continue to operate.
Mr Varadkar said: "Above all, we all need to look out for each other.
"Ireland is a great nation. And we are great people.
"We have experienced hardship and struggle before. We have overcome many trials in the past with our determination and our spirit.
"We will prevail."
EARLIER: Taoiseach Leo Varadkar has announced that all schools, colleges and childcare facilities in Ireland are to close, from 6pm today.
Speaking in Washington, Mr Varadkar said cultural institutions will also close but public transport will continue to operate.
While people are to continue to work where possible, they are asked to work from home if they can and stagger start times and breaks to limit contact.
Shops, restaurants and other businesses are to remain open but limitations are being placed on indoor and outdoor gatherings of people.
Indoor gatherings of 100 people or more are cancelled, while outdoor gatherings of 500 or more are also cancelled.
The Taoiseach said: "This will involve big changes and enormous sacrifices but we are doing it for each other. Economy will bounce back, lost time at school can be recovered, we must look out for each other.
"Where possible work from home. Public transport will continue to operate. Restaurants can stay open but must operate social distancing.
"The cabinet will meet later today."
These measures will remain in place until at least March 29.
Speaking to the public, Mr Varadkar said people can play your part by hand washing coughing and sneezing and your elbow or tissue and seeking medical advice if you develop symptoms.
“This is now more important than ever,” he said.
Schools across Cork had been preparing to close amid the current coronavirus outbreak in a bid to prevent further spread.
Teachers have told The Echo that they are preparing systems to teach and grade students from home over the coming weeks.
Speaking to The Echo, Coláiste Éamann Rís principal Aaron Wolfe described the current situation for schools as “unprecedented”.
He said the school has been preparing for the possibility of forced closure amid increased cases of the virus Covid-19, which has killed one person in Ireland and thousands worldwide.
“We have been preparing with Google classes and the students will be able to access webinars and other resources from home,” explained Mr Wolfe.
“It’s an unprecedented situation but our staff have been working hard to make sure they can teach and grade students from home."
More to follow.