"We're experiencing a global system outage," the airline said in reply to one disgruntled traveller on Twitter.
BA has apologised to customers as scores of Britons headed overseas for the long weekend and half-term school holidays on Saturday morning.
A spokeswoman said: "We apologise to customers who are facing some delays following an IT outage this morning.
"We are working to resolve the problem as quickly as possible."
The airline has experienced problems with its online check-in systems in the past.
Passengers were hit by severe delays in September and July last year because of IT glitches.
The latest problem meant parts of BA's website were unavailable and some travellers claimed they could not check in on the mobile app, with many venting their frustration on social media.
One passenger, Henry Tail, tweeted: "Hey @British_Airways, couldn't log in to the app with my boarding pass on because systems were down and I missed my flight. What can I do?"
Another, Nadia Whittley, posted: "British airways computers down T5 and standstill, stranded on the plane and they make us pay for food!! Please retweet this indicency! (SIC)"
Julie Adie wrote: "We are on the runway.. For hour now..no offer of drinks..and because ba only takes cards now, we only have cash...wat we supposed to do!!?? (SIC)"
Melissa Davis, who runs MD Communications, a legal PR agency in London, was held for more than an hour and a half on the tarmac at Heathrow, on board a BA flight returning from Belfast.
Speaking from the plane, Miss Davis said the air conditioning had been off "so I don't think we will be going anywhere any time soon", but added that the passengers had been kept informed by their pilot and given water while they remained seated.