The casualties were taken to hospital after the accident at the Lecht Ski Centre in the Cairngorms, but their injuries are not believed to be life-threatening.
Another 36 skiers were stranded on one of the chairlifts and were rescued by Grampian Fire and Rescue Service.
The incident occurred yesterday afternoon during one of the busiest times of the year, after families flocked to the slopes during the mid-term break for schools.
Scotland's ski centres have an almost unblemished record for safety, but at about 3.25pm the chairlift dropped to the ground when the tow line came off the wheels about halfway up a hill next to the main complex.
A Royal Navy search and rescue helicopter was dispatched from HMS Gannet at Prestwick Airport to the snowsports facility in Aberdeenshire.
Fire and rescue units were joined by Scottish Ambulance Service personnel, Grampian Police and Braemar mountain rescue teams. One rescuer said the patients had been calm despite their ordeal.
A spokeswoman for the Lecht operation said: "All casualties are in a stable condition, three were taken to hospital by helicopter and three by ambulance.
"Staff evacuated 36 people who were stuck on the other chairlifts and everyone was off the slope within an hour and 40 minutes.
"We don't know the reason for the accident, but the Health and Safety Executive and the Lecht will be investigating the cause."
Grampian Police said everyone who was stranded was checked over by medical personnel at the ski centre. At least two of the injured were taken to Aberdeen Royal Infirmary for treatment.
Richard Lochhead, the MSP for Moray said: "This is a dreadful accident at what is a very popular destination for families. I am sorry to hear about people being injured on what should have been a fun day out and it must have been a horrific experience for those trapped on the ski lift, especially the children."
Moray councillor Michael McConachie, who represents Speyside Glenlivet, added: "I sincerely hope everyone involved is okay. My heartfelt thoughts go out to the family and friends of everyone involved."
The Lecht is on the A939 road between Cockbridge and Tomintoul on the eastern side of the Cairngorms.
It was opened in 1977 by Ronnie Winram and Professor Jim Petrie as skiing's popularity grew in the UK. At first the centre was little more than a wooden hut. Nowadays it has the capacity to uplift more than 10,000 skiers an hour. The site has a total of 12 different lifts, including a three-man chairlift and a magic carpet travelator for beginners.
In July 2006, five people including a three-year-old girl were injured when a cable car plunged plunged 60ft to the ground after colliding with another gondola at Aonach Mor in the Nevis Range resort.