In order to grade assignments go to the Administration block and click "Grades". Here you will see all of your students for that class, the Categories you set up at the beginning of class, and the Assignments under each category. In order to grade an assignment click on the link at the top of the column with the assignment's name.
This brings up a Submissions page for that assignment where you can view what students sent you by clicking on the link under the "Last Modified" column. To grade the submission click on the link for "Grade". This brings up a new window with a "Grade" drop-down menu, an HTML window for providing feedback to the student, a window for the student to respond to the grade, and the original submission at the bottom.
You can "Save changes", which will close the window, as will the "Cancel" button, or you can "Save and show next", which will allow you to grade the next student easily.
The "Send notification emails" will allow students to be informed when their grades are ready.
When you're finished the Submissions page will reflect the changes you have made. You can still change grades here by clicking the "Update" link under Status.
Grading Assignments with Uploaded Files
To grade assignments involving uploaded files click on the "Grades" link and then the link to the assignment at the top of the appropriate column. This will take you to the same "Submissions" page as before. In the "Last Modified" column, instead of a link to the grading window you will see a link to the uploaded file. To view the document click on its link and the browser will download it. Then you can grade the assignment and leave whatever comments you want.
Grading Offline Activity
Go to the "Submissions" page for the assignment. You will see that there is nothing submitted. You can enter in grades as before.