I saw a ship a'sailing, a'sailing on the sea...

Harbor Scene: An English Ship with Sails Loosened Firing a Gun
Peter Monamy

One of the ways to find an immigrant ancestor is to locate them as a passenger on a ship traveling to their new country. One of the first places to go for passenger lists to the United States and elsewhere is Ancestry.com. Presently, Ancestry.com has 194 collections of passenger lists with well over 100 million records. Here is a screenshot of the list from the Ancestry.com Card Catalog;

Some time ago, Lorine McGinnis Shulze listed many of the major sources in her post on The Olive Tree Genealogy entitled appropriately, "Ships Passenger Lists." Cyndi Howells on her Cyndi's List also has extensive links to passenger list web sites. See Ships & Passenger Lists. There is also a helpful list of resources on U.S. National Archives website for Immigration Records (Ship Passenger Arrival Records). You may also wish to try the One-Step Webpages by Stephen P. Morse.

I have also found that searching on Google for the name of the ship, if known, can be very useful. If you do not know the name of the ship, just try searching with Google for the topic "Passenger Lists" and you will find many more resources.