In order for a user to edit content on the website, they must have the Page Manage role. To assign them that role, you must go to Core:
  1. Select Security > Role
  2. Click on Page Manager
  3. Click on Members
  4. Click Add
  5. Search for the user by Last Name
  6. Click the blue arrows to the right of their name to move them to added users
  7. Click Save & Exit

If you do not want to give a user the Page Manager role, but still want them to be able to edit website content, follow the steps in How can I give a user access to edit certain onMessage site pages?