I have a few general questions about modifying Magento’s admin section and would be grateful to have them answered. I’m new to Magento so please bear with me.
My goal is to add a new column with a product attribute (e.g. “Size”) to the “Category Products” table within the Catalog -> Manage Cateories section (see screenshot below).
Having little Magento dev experience, I’m not quite sure where to start. I had a look in some of the Grid.php files under the adminhtml directory, and while I see a bunch of statements like addColumn(...), I’m not sure where I’d slot in my new attribute column.
Also, I assume that instead of modifying any core files directly, I’d copy them to the same path under the local folder and edit or somehow extend them there? Do I have to edit any config files or do anything else for the change to be reflected? Am I – by doing this – in effect creating my own module?
I also read that I should disable “Compilation” before I make any changes. Why is this? Is there anything else to consider?
Again I am very grateful for any help and appreciate that my questions must seem basic. Any supplementary resources you could point me towards would be appreciated. Thanks.

Max solution was pretty spot on but missing some important steps, I’ll elaborate on his original method
Create a new local override of the Product Tab by copying app/code/core/Mage/Adminhtml/Block/Catalog/Category/Tab/Product.php to app/code/local/Mage/Adminhtml/Block/Catalog/Category/Tab/Product.php
There are 2 functions involved in modifying the grid view. _prepareCollection and _prepareColumns
_prepareColumns by adding a call to the addColumn function just like:
_prepareCollection, by default the product collection loaded in the grid only has a few attributes(name,sku,price) what you need to do add our now attribute by ->addAttributeToSelect(‘size’) now if you are only working with a textfield attribute then this is the extend of the modifications you have to do however if your attribute is for example a dropdown you will need to do further changes to the prepare collection:
(optional) dropdown attributes only store the value of the option that was select so we need to provide an options array to the addColumns call so Magento can display the values correctly, we can do that in the following maner:
on your local copy of Products, add the following to the _prepareColumns functions