Gallery has lots of places module authors can add action links and tabs. Here's a brief overview of some of them, with pointers to canonical (core code) examples that you can copy and modify when writing your own modules.
SystemLinks appear at the top of every Gallery Page. They're declared in module.inc, in function getSystemLinks().
ItemLinks are the actions that apply to each item. In Matrix Theme (the default) the ItemLinks for the current item usually appear in the left sidebar. For album views there's a dropdown or list of ItemLinks for each item in the album too. ItemLinks are declared by modules also in main.inc, in function getItemLinks(...). Your module can distinguish between the main item for the page which has $wantsDetailedLinks set to true for that item, and contained items. See /modules/core/module.inc, in class CoreModule for examples.
ItemAdminViews are things that do administrative actions on items. Mostly they are also declared as ItemLinks too. ItemAdminViews are implemented as subviews of the core.ItemAdmin view so the declaration of the View and its corresponding Controller and Template is somewhat different to normal, as some of the work (form declarations etc) are done by the main ItemAdmin View/Template/Controller. In the example above you are looking at the RearrangeItems.tpl template from the 'rearrange' module, displayed as 'sub-template' of the core.ItemEdit template.