From the Minnesota Extension Service, an ice dam is:
"a ridge of ice that forms at the edge of a roof and prevents melting snow (water) from draining off the roof. The water that backs up behind the dam can leak into a home and cause damage to walls, ceilings, insulation and other areas."
A thick row of icicles hanging from your eves indicates that you have an ice dam on your roof. Right now we've entered the time of year when the temps can reach above freezing in the day and dip below freezing over night. This will only make existing ice dams worse, as the snow on your roof melts in the day, encounters the dam, has nowhere to go, and freezes into an even thicker dam over night.
When the water starts flowing indoors, you run the risk of having to replace insulation, sheet rock, carpeting, and whatever happened to be sitting below the place where the leak happened. The cost of repairing the damage caused by an ice dam leak far exceeds the cost of preventative maintenance!
Home owners can prevent ice dams from forming in a few ways: Melting cables can be installed along the eve, snow can be raked from the eves using a special roof rake, or homeowners can have their roof professionally cleaned. Once ice forms dam, ice dam removing professionals can steam away the icy buildup before the condition gets critical. Need help finding a qualified contractor? See our post on how to vet contractors!
For more detailed information on ice dams and ice dam prevention, read the article: Minnesota Extension Service, Ice Dams