I am a new MS Access user. I am having difficulty adding columns to a pre-existing database. Is there a simple way to do this?

If I understand your question correctly, you would open the table in design view and add new fields. Access doesn't have columns (even though when the table is open, it looks like a spreadsheet).

Access does have columns. It is cells it doesn't have. Column is another name for Field.
Field/Record/File -- terminology that's been around since the beginning.
Column/Row/Table -- terminology found in RDBMS literature
Attribute/Tupple/Relation -- terminology found in academic literature.

Ok,
so far so good. But now I can only open Design View as read-only so the changes are not saved. Can I change my User privileges or break the password (if there is one)? When I try to open in Design View I get the following message:
"You can't open the table 'name' for modification. A query or form bound to the table is open, you may not have permission to open this table in Design view, or another user has the table open. Do you want to open this table as read-only?"

You can't be making modifications to objects while someone else has the database open! To make changes, proper procedure is to copy the front end to your local drive. Copy the back end to your local drive. Make the necessary changes. Replace the shared copy of the front end on the server. And be thankful you don't work for a company that maintains strict quality control over software. One of my clients has EIGHT levels in their release to production process. Each time the application is tested and approved at one level, it is copied to the next where it is tested some more until it gets to the final staging library. After it is approved there, it is moved to the shared production library.