The tracking data that Magento stores is very limited. Beyond the tracking number and the name of the carrier there's not a lot to see but if this is data data you want you would go about that something like this;
The tracking data is stored in the table
sales_flat_shipment_track so using the appropriate resource model you can filter the collection by the given tracking number.
This will return a collection from which you can retrieve the first item and the data of that tracking item. (untested code, please test first)
$tracking number = 'ABCDEFG'; // input from user
$tracking_collection = Mage::getResourceModel('sales/order_shipment_track')
->addAttributeTofilter('track_number', $tracking number);
echo "No order found";
$tracking_item = $collection->getFirstItem();
Now for the part where you provide a user a form to retrieve the data, this can be done by adding a simple phtml file with a form in the
<block type="core/template" name="tracking_info" template="tracker/form.phtml"/>
Actually letting the form submit something to the above mentioned code and returning something will probably involve creating an extension with a controller that handles the post and returns the info. This Magento-4u tutorial should help you with that.
Now if you meant retrieving tracking data from the respective carrier (postal service) this will depend completely on what carrier you're using and might be a bit too broad to go into in this post.