What others have said in a thousand words: The failure of the gulf stream current, which warms Europe, UK and the east coast of north America, could occur because of a large input of fresh water into the north Atlantic. Quick version is it reduces the salinity of the north Atlantic so the water stops sinking and the deep sea current carrying cold water south to the equator slows or stops flowing.
This could occur due a melt of any of the large glaciers in Russia as well as Greenland but potentially could also occur due to the melting of a large part of the Antarctic ice pack as this would effect the deep water currents in the southern Atlantic and pacific. All the ocean currents carrying warm water on the surface and all the deep sea currents carrying cold water in the depths are connected and what happens to one will effect the others.
The first sign of the failure of the currents would be that the equatorial regions of the Atlantic and pacific would get hotter,this is because the deep sea currents carrying cold water to the equator would fail first, causing the waters of the equator to warm up. This could potentially cause an increase in storm activity around the equator long before the warm water currents moving north started to slow.
Eventually both sets of current would fail, at this point the temperatures in those areas of the north Atlantic currently warmed by the gulf stream would begin to cool down. Contrary to what you may have seen in 'The day after tomorrow,' the reduction in temperatures would probably be gradual and limited to just a few degrees. Unfortunately this would be combined with an increase in storms because while the extreme north and south are getting colder, the equator is getting hotter and nature hates a difference in temperatures like that. You would still get the same exchange of cold water, from the arctic regions, for warm water, from the equator, occurring but instead of the transfer occurring in the oceans it would occur in the atmosphere. Interaction between the colder and drier atmosphere in the northern north Atlantic and the warmer and wetter atmosphere in the southern north Atlantic would generate storm systems which would travel north. Also with the lower temperatures the precipitation from these storms would be more likely to fall as snow, particularly at higher altitudes and in the extreme north.
Gradually over decades glaciers would form, and existing glaciers would grow, at upper altitudes as snow survives from one winter to the next and is compressed by fresh snowfall into ice. This would reflect sunlight back into space causing temperatures locally to fall further and causing more precipitation to fall as snow and for that snow to survive from one winter to the next at lower and lower altitudes.
This is how a failure of the deep sea currents could trigger a new ice age.