Can an abstract class have a constructor?
If so, how can it be used and for what purposes?
Not only can it, it always does. If you do not specify one then it has a default no arg constructor, just like any other class. In fact, ALL classes, including nested and anonymous classes, will get a default constructor if one is not specified (in the case of anonymous classes it is impossible to specify one, so you will always get the default constructor).
A good example of an abstract class having a constructor is the Calendar class. You get a Calendar object by calling Calendar.getInstance(), but it also has constructors which are protected. The reason its constructors are protected is so that only its subclasses can call them (or classes in the same package, but since it's abstract, that doesn't apply). GregorianCalendar is an example of a class that extends Calendar.