This sounds like your players should be products and your teams the categories. Then you can assign the products/players to multiple categories/teams.
That said, to more directly answer your question, here's an Inchoo Article detailing how to add attributes to categories, hopefully you can use it as a starting point if you want/need to go that route.
Copying from the article in case it goes missing later:
SQL setup script:
$installer = $this;
$installer->startSetup();
$entityTypeId = $installer->getEntityTypeId('catalog_category');
$attributeSetId = $installer->getDefaultAttributeSetId($entityTypeId);
$attributeGroupId = $installer->getDefaultAttributeGroupId($entityTypeId, $attributeSetId);
$installer->addAttribute('catalog_category', 'new_cat_attrb', array(
'type' => 'int',
'label' => 'New Category Attribute',
'input' => 'text',
'global' => Mage_Catalog_Model_Resource_Eav_Attribute::SCOPE_STORE,
'visible' => true,
'required' => false,
'user_defined' => false,
'default' => 0
));
$installer->addAttributeToGroup(
$entityTypeId,
$attributeSetId,
$attributeGroupId,
'new_cat_attrb',
'11' //last Magento's attribute position in General tab is 10
);
$attributeId = $installer->getAttributeId($entityTypeId, 'new_cat_attrb');
$installer->run("
INSERT INTO `{$installer->getTable('catalog_category_entity_int')}`
(`entity_type_id`, `attribute_id`, `entity_id`, `value`)
SELECT '{$entityTypeId}', '{$attributeId}', `entity_id`, '1'
FROM `{$installer->getTable('catalog_category_entity')}`;
");
//this will set data of your custom attribute for root category
Mage::getModel('catalog/category')
->load(1)
->setImportedCatId(0)
->setInitialSetupFlag(true)
->save();
//this will set data of your custom attribute for default category
Mage::getModel('catalog/category')
->load(2)
->setImportedCatId(0)
->setInitialSetupFlag(true)
->save();
$installer->endSetup();
And the config.xml:
<resources>
<new_attribute_csv_setup>
<setup>
<module>New_Attribute</module>
<class>Mage_Catalog_Model_Resource_Eav_Mysql4_Setup</class>
</setup>
<connection>
<use>core_setup</use>
</connection>
</new_attribute_setup>
<new_attribute_setup_write>
<connection>
<use>core_write</use>
</connection>
</new_attribute_setup_write>
<new_attribute_setup_read>
<connection>
<use>core_read</use>
</connection>
</new_attribute_setup_read>
</resources>