You can add new sections, new fields and customise the public visibility of existing field on a users profile. To edit user profiles click on the Actions button > Configure, and then select Customise User Fields under Users.
You will then be presented with edit mode for the profile page along with the Pencil Toolbox that allows you to drag new fields and new sections into the profile. You can then use the three dots for each field and decide whether it can be seen publicly and who can edit it amongst other options. Different fields may offer slightly different options depending on which tool you have used from the pencil icon.