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array(
    '_POST' => (array) $_POST,
    'newCustomer' => (object) Customer object
);

array(
    'carrier' => (object) Carrier object,
    'order' => (object) Order object
);

array(
    'orderReturn' => (object) OrderReturn object
);

array(
    'order' => Order oject,
    'productList' => array(
        (int) product ID 1,
        (int) product ID 2,
        ...,
        (int) product ID n
    ),
    'qtyList' => array(
        (int) quantity 1,
        (int) quantity 2,
        ...,
        (int) quantity n 

    )
);

The order of IDs and quantities is important!

array(
    'newOrderStatus' => (object) OrderState object,
    'id_order' => (int) Order ID
);

array(
    'newOrderStatus' => (object) OrderState object,
    'id_order' => (int) Order ID
);

array(
    'paymentCC' => (object) OrderPayment object
);

array(
    'id_order' => (int) Order ID
);

After the search in the store. Includes both instant and normal search.

array(
    'expr' => (string) Search query,
    'total' => (int) Amount of search results
);

array(
    'old_id_shop' => (int) Old shop ID,
    'new_id_shop' => (int) New shop ID
);

array(
    'id_product' => (int) Product ID,
    'id_product_attribute' => (int) Product attribute ID,
    'quantity' => (int) New product quantity
);

array(
    'cart' => (object) Cart object,
    'order' => (object) Order object,
    'customer' => (object) Customer object,
    'currency' => (object) Currency object,
    'orderStatus' => (object) OrderState object
);

array(
    'id_image' => (int) Image ID,
    'id_product' => (int) Product ID
);

array(
    'id_customer' = (int) Customer ID
); 

array(
     'id_order' = (int) Order ID
); 

array(
    'carriers' => array(
        array(
            'name' => (string) Name,
            'img' => (string) Image URL,
            'delay' => (string) Delay text,
            'price' =>  (float) Total price with tax,
            'price_tax_exc' => (float) Total price without tax,
            'id_carrier' => (int) intified option delivery identifier,
            'id_module' => (int) Module ID
    )),
    'checked' => (int) intified selected carriers,
    'delivery_option_list' => array(array(
        0 => array( // First address
            '12,' => array( // First delivery option available for this address
                 carrier_list => array(
                     12 => array( // First carrier for this option
                         'instance' => Carrier Object,
                         'logo' => <url to the carrier's logo>,
                         'price_with_tax' => 12.4, // Example
                         'price_without_tax' => 12.4, // Example
                         'package_list' => array(
                             1, // Example
                             3, // Example
                          ),
                     ),
                 ),
                 is_best_grade => true, // Does this option have the biggest grade (quick shipping) for this shipping address
                 is_best_price => true, // Does this option have the lower price for this shipping address
                 unique_carrier => true, // Does this option use a unique carrier
                 total_price_with_tax => 12.5,
                 total_price_without_tax => 12.5,
                 position => 5, // Average of the carrier position
             ),
         ),
     )),
     'delivery_option' => array(
         '<id_address>' => Delivery option,
         ...
     )
); 

NOTE: intified means an array of integers 'intified' by Cart::intifier

array(
    'module' => (string) Module link [e.g. /module/test/front]
);

array(
    'address' => (object) Address object
);

array(
    'total_to_pay' => (float) Total amount with tax,
    'currency' => (string) Currency sign,
    'objOrder' => (object) Order object,
    'currencyObj' => (object) Currency object
);

array(
    'order' => (object) Order object
);

array(
    'total_to_pay' => (float) Total amount with tax,
    'currency' => (string) Currency sign,
    'objOrder' => (object) Order object,
    'currencyObj' => (object) Currency object
);

array(
    'product' => (object) Product object
);

array(
    'cart' => (object) Cart object
);

Note that the Cart object can also be retrieved from the current Context.

displayHeader

A hook which allow you to do things in the header of each pages

displayHome

displayInvoice

Add blocks to invoice (order)

displayLeftColumn

displayLeftColumnProduct

displayMyAccountBlock

Display extra information inside the "my account" block

displayMyAccountBlockfooter

Display extra information inside the "my account" block

displayOrderConfirmation

Called on order confirmation page

displayOrderDetail

Displayed on order detail on front office

displayPDFInvoice

Allow the display of extra information into the PDF invoice

displayPayment

displayPaymentReturn

displayPaymentTop

Top of payment page

displayProductButtons

Put new action buttons on product page

displayProductComparison

Extra Product Comparison

displayProductTab

Called on order product page tabs

displayProductTabContent

Called on order product page tabs

displayRightColumn

displayRightColumnProduct

displayShoppingCart

Display some specific information

displayShoppingCartFooter

Display some specific information on the shopping cart page

displayTop

array(
    'delivery' => $delivery,
    'delivery_state' => State::getNameById($delivery->id_state),
    'invoice' => $invoice,
    'invoice_state' => State::getNameById($invoice->id_state),
    'formattedAddresses' => $formatted_addresses,
    'products' => array_values($products),
    'gift_products' => $gift_products,
    'discounts' => $cart_rules,
    'is_virtual_cart' => (int)$this->isVirtualCart(),
    'total_discounts' => $total_discounts,
    'total_discounts_tax_exc' => $total_discounts_tax_exc,
    'total_wrapping' => $this->getOrderTotal(true, Cart::ONLY_WRAPPING),
    'total_wrapping_tax_exc' => $this->getOrderTotal(false, Cart::ONLY_WRAPPING),
    'total_shipping' => $total_shipping,
    'total_shipping_tax_exc' => $total_shipping_tax_exc,
    'total_products_wt' => $total_products_wt,
    'total_products' => $total_products,
    'total_price' => $base_total_tax_inc,
    'total_tax' => $total_tax,
    'total_price_without_tax' => $base_total_tax_exc,
    'is_multi_address_delivery' => $this->isMultiAddressDelivery() || ((int)Tools::getValue('multi-shipping') == 1),
    'free_ship' => $total_shipping ? 0 : 1,
    'carrier' => new Carrier($this->id_carrier, $id_lang),
);

Can be found in /classes/Cart.php

array(
    'delivery' => $delivery,
    'delivery_state' => State::getNameById($delivery->id_state),
    'invoice' => $invoice,
    'invoice_state' => State::getNameById($invoice->id_state),
    'formattedAddresses' => $formatted_addresses,
    'products' => array_values($products),
    'gift_products' => $gift_products,
    'discounts' => $cart_rules,
    'is_virtual_cart' => (int)$this->isVirtualCart(),
    'total_discounts' => $total_discounts,
    'total_discounts_tax_exc' => $total_discounts_tax_exc,
    'total_wrapping' => $this->getOrderTotal(true, Cart::ONLY_WRAPPING),
    'total_wrapping_tax_exc' => $this->getOrderTotal(false, Cart::ONLY_WRAPPING),
    'total_shipping' => $total_shipping,
    'total_shipping_tax_exc' => $total_shipping_tax_exc,
    'total_products_wt' => $total_products_wt,
    'total_products' => $total_products,
    'total_price' => $base_total_tax_inc,
    'total_tax' => $total_tax,
    'total_price_without_tax' => $base_total_tax_exc,
    'is_multi_address_delivery' => $this->isMultiAddressDelivery() || ((int)Tools::getValue('multi-shipping') == 1),
    'free_ship' => $total_shipping ? 0 : 1,
    'carrier' => new Carrier($this->id_carrier, $id_lang),
);

Can be found in /classes/Cart.php