What we mean by ‘write access’ is that some ESI scopes can effect change on Eve, such as some endpoints setting your autopilot destination, allowing others to send eve-mails in your name, or adding contacts and setting their standings. Read only access would let people see what they need to see, but not to change anything or initiate an action for you.
You can check the available scopes at https://esi.evetech.net/ui/. Scopes seem to be color coded, at a hasty glance, by their descriptions on the right hand side when expanded. Blue are read only, green allows for adding things, yellow modifications, and red deletion. Get methods are generally read. Post and Put are generally write, but, like the pirate code, it seems to be more a guideline than an actual rule.
Also, I do not have a thorough understanding of how these web methods work, so take it as the leyman’s ‘in a nutshell’ oversimplified version of a complex subject.