In order for a fat molecule to be saturated, it must have what characteristic?

Fat (lipid) molecules are made up mostly of long chains of carbon atoms.
For a fat molecule to be considered saturated, the carbon chain must have only single bonds between them. 
Schematically, they look like  -C-C-C-C-  ...

If the carbon chain contains one or more double bonds, such as  -C=C-C-C, then the given lipid molecule is unsaturated.

If a lipid molecule (fatty acid) contains exactly one double bond in its chain, and the rest are single bonds, it is then called monosaturated.  It is a more healthy food source than saturated fats.

Saturated fat are generally solid at room temperatures.  Examples include lard, butter, mostly from animal sources.  However, coconut oil (solid at room temperatures as well) is a saturated fat.