This immigration database for Galveston, Texan is brought to us by the Galveston Historical Foundation. The data may be offered by other sources, but having multiple places to look for the same information is a luxury.
From the landing page, click on the link "Access the Online Database." You can search by name, year of arrival, and ship's name. The hit list gives simply index information, but has a "details" link with more information from the individual records.