The Blackboard "Assignment" tool provides a way for you to give instructions and attached files to a students, a "dropbox" for the student to turn in files (papers, documents, spreadsheets, etc), and then a means for you to grade, give feedback and return files.
Note that an "Assignment" is what you would use for an assessment where you expect the student to submit some type of file. For a reading assignment, or when you just need to give instructions you would create an Item. Assignments are also one-to-one between the student and instructor - if you want students to view and comment on each others work then a Discussion forum might be more appropriate.
Online Tutorial: About Blackboard Assignments
Collecting and Grading Assignments
Assignment files are viewed by the instructor through the Grade Center. That is where you will view or download files, enter grades, feedback and any returned files. The student will go to their own gradebook (My Grades) to see their score and when they click on that they will be able to view feedback and receive any returned files.
You can also view this Elluminate (WebMeeting) Recording