Footage on social media showed thousands of people queueing outside Terminal 2 on Saturday morning at the airport after digital systems failed.
The airline said its systems had now been restored and flights on Sunday were scheduled to operate as normal.
The airline said that customers affected by the disruption will be able to change their travel plans free of charge.
A company statement said: “Aer Lingus sincerely apologises to customers for the severe disruption caused today by the unavailability of key systems for check-in, boarding and our website.
“This system outage was caused by a break in connectivity in services from a UK network provider.
“The break in connectivity has now been restored and we are taking steps to bring all of our systems back online.
“Aer Lingus had to cancel 51 flights today, mainly to and from Dublin/European and UK destinations.”
The airline said it is operating all of its transatlantic services from Dublin, albeit with delays and reduced passenger numbers in some instances due to security restrictions applying as a result of the systems outage.
It said transatlantic services from Shannon operated as normal and Aer Lingus Regional (operated by Emerald Airlines) services would operate with some delays.
Aer Lingus advises customers that its operations for tomorrow are planned to operate as normal
The statement said: “Any customer impacted by today’s disruptions will be able apply for a refund or change their travel plans, free of charge through Aerlingus.com, our call centres and our social media channels.
“As systems have now been restored we are contacting customers directly in order to re-accommodate them as efficiently as possible.
“We have made additional customer service agents available to deal with high call volumes – please bear with us as we try to service all customer queries.
“We will also share information regarding customers’ rights.
“Aer Lingus advises customers that its operations for tomorrow are planned to operate as normal.
“Customers are advised to come to the airports at the normal time for their scheduled flight.”
The technical failure affected check-in, boarding and the company website.
It led to the airline having to cancel most flights to and from Dublin Airport to Europe and the UK after 2pm.
It also left the airline unable to communicate directly with customers, many of whom turned up at Dublin airport.
In a tweet, Dublin Airport said impacted passengers should contact the airline directly about next steps.
The airport also said that no other airlines are affected.
Customers reported long queues at the airport on Saturday as Aer Lingus check in for earlier flights was being handled manually.
Due to ongoing @AerLingus IT issues, the airline has confirmed the cancellation of all @AerLingus flights to the UK & Europe that were due to depart Dublin Airport after 2pm today (Saturday). Impacted passengers should contact @AerLingus regarding next steps. pic.twitter.com/XnZLgpCFpB
— Dublin Airport (@DublinAirport) September 10, 2022
Lisa Webb from Which? said: “Aer Lingus’ sudden cancellations of flights to and from the UK and Europe will cause thousands of passengers distress and anxiety when it comes to their travel plans.
“In events such as these, airlines should be offering the option of a refund or to reroute passengers on any reasonable route as quickly as possible, using other airlines where necessary.
“Aer Lingus must also give clear information to affected customers about their entitlement to compensation.”