The assisgments function is not new to Blackboard but it has been improved and it's now used in place of the digital drop box function (to submit assigments online). The links below will help you as you integrate assignments into your instruction.
Getting Started with Assignments (.pdf) - Updated Item
Create an Assignment in Blackboard
Downloading Assignments (to work offline)
Viewing Assignments after Grading
Clean Out Graded Assignments File
Accessing a SafeAssign Assignment