What makes water condense on the interior of the windows at night when it's cold?
There are two things working together to make water condense on the windows.  The house is humid, and the windows are cold (even well-insulated windows will usually be the coldest thing in the house because of the low R-value compared to walls and ceilings).  To prevent the condensation, you can remove the humidity or make the windows warmer.

Removing Humidity

You have several options for removing humidity.


Ventilation system

Running exhaust fans in areas that produce humidity (like the bathroom while showering) will lower the overall humidity in the house.
Forced air heating

Forced air heating systems wick moisture from the air.  If you have a forced air system at your house, and your relatives have radiated heat from hot-water baseboards, this may help explain why they have condensation on their windows and you do not.
Dehumidifier

Dehumidifiers live up to their name.  This one claims it can remove 70 pints of water from a 1400sqft house every day.  That's almost 9 gallons of water taken out of the air each day.  Just remember to put it near a drain.


Making the Windows Warmer

The other option is to make the windows warm enough that the water does not condense.


Plastic

Placing a sheet of plastic over the inside of the windows will insulate the windows better.  If the plastic seals all the way around the window frame, it also prevents the humid air from reaching the window in the first place.
Space Heater

If there are only a couple of windows with serious condensation issues, a space heater directed toward those windows should heat the windows enough to lessen or eliminate the condensation.