How do I enable editing the body of a block?
You are facing this issue because you probably created the block as a user who has access to a text format like Full HTML but the other user doesn't have that permission :( Here's what you can do:


Sign in as the root user (uid 1) who created the block;
Change the text format to Basic HTML or Restricted HTML.
Go to Admin > People > Permissions page and make sure the other non-admin user has the permissions to use the text format you applied.
Depending on your requirements, you can ignore point 2 and give Full HTML permissions to the concerned user.