See how to add a new user to a SharePoint list so he / she has access it and how to remove a user from a SharePoint list with this end user SharePoint Tutorial.
Lists and Libraries are part of a site and inherit permissions from the site. If you have author permission at site level you can also change settings of a list or library and you can delete its contents (as long as the default permissions haven't been changed). Stopping the inheritance of permission is a common task if you store important information in that list.
If you want to manage permissions of a list or library you have to be the site owner or you need to have the "manage permissions" permission. After stopping the inheritance you can change permission for users or groups.
How to add users to a SharePoint list or remove them
If you want to add or remove users from a SharePoint list open the list and click the list tab. Go to the list permissions and stop the permission inheritance. Add or remove users as you want.