Passengers will still be able to travel with two items of hand luggage, but the larger bag will be placed in the hold unless they pay for priority boarding.
Only the smaller bag - which can be the size of a handbag or laptop bag - will be allowed in the cabin.
The policy means many more travellers will have to wait at airport carousels to collect bags on arrival.
Priority boarding costs from £5.
Ryanair has also lowered the price of flying with a check-in bag from £35 to £25 and increased the weight allowance from 15kg to 20kg.
Chief marketing officer Kenny Jacobs said the change will reduce delays caused by passengers struggling to fit wheelie bags into overhead lockers and will cost the Dublin-based carrier up to 50 million euro (£45 million) each year.
He went on: "We hope that all our customers will enjoy these new bag rules which are centred around lower checked bag fees and bigger bag allowances.
"All Ryanair customers will still be free to bring two free carry-on bags but because of our heavily booked flights we don't have space on board for this many wheelie bags."