You probably want the custom field to be on the product page instead of on the order page in case your customers buy multiple t-shirts.
Give this video a watch: http://www.magentocommerce.com/magento-on-the-fly/custom-product-options/
It shows how you can add a custom option (like a text field or text area) where your customers could type in the phrase they want to be on the shirts, and that will get pushed through in the order process.
If that doesn’t work for you, there are a few commercial extensions in Magento Connect that allow you to add custom fields to the checkout process itself. I have not tested these and don’t know how well they work.
Hope it helps. Good luck!