Determining the number of days from today
Let's start with the main error that I see: You compute the timespan between the "current date" and the "due date". If the current date is today at 11am, and the due date is tomorrow at 10am then the difference
is 0 days and 23 hours, to this is reported as "today".

This is probably not what you wanted, therefore todayDate must be
set to the start of the current day. This is easily done with

let todayDate = calendar.startOfDayForDate(NSDate())


Then todayDate and calendar should be computed only once before the
loop starts, and not for each loop iteration.

There are also problems in the analysis of the computed date components:

if calendarComponents.month &gt; 0 &amp;&amp; calendarComponents.day &gt; 0 { ...


What if the difference is exactly 2 (or more) month and 0 days?

} else if calendarComponents.day &gt; 7 {


What if the difference is exactly one month and 0 days?

The first two if-conditions should therefore be

if calendarComponents.month &gt;= 2 || (calendarComponents.month == 1 &amp;&amp; calendarComponents.day &gt; 0) {
    // More than a month
} else if calendarComponents.month == 1 || calendarComponents.day &gt; 7 {
    // Between 8 days and one month
} else if ...


Other things that can be improved:

If you create a NSManagedObject subclass for your entity then
you can use the property accessors instead of valueForKey():

let fetchedResults = managedContext.executeFetchRequest(fetchRequest,
    error: &amp;error) as! [Assignment]?

// ... later ...
let dueDate = item.dueDate


The advantage is that you don't need the cast as! NSDate anymore
and the compiler can check for correct types.

Some type annotations are not necessary, e.g.

let calendarComponents: NSDateComponents = calendar.components(NSCalendarUnit.CalendarUnitYear | NSCalendarUnit.CalendarUnitMonth | NSCalendarUnit.CalendarUnitDay,
    fromDate: todayDate, toDate: dueDate, options: .allZeros)


can be simplified to

let calendarComponents = calendar.components(.CalendarUnitYear | .CalendarUnitMonth | .CalendarUnitDay,
    fromDate: todayDate, toDate: dueDate, options: nil)


And 

println("Days: " + calendarComponents.day.description + " Months: " + calendarComponents.month.description) 


can be simplified with string interpolation:

println("Days: \(calendarComponents.day) Months: \(calendarComponents.month)")