Whether creating transparency in a file for the Web or print (or even video), you'll want to use selection to define the part of the image you want to keep. Once the selection is created, you can use that selection to create the image elements you'll need to create transparency.
You have to make the selection only once if you do it right; you then can save the image to multiple formats, and use it both in print and on the Web. The details of selection can be very involved. To focus on the basic process rather than creating selection (which is at least one other article), I assume that you know how to use some of the basic tools to make the selection. I am also leaving out the discussion of the way resolution affects the changes, except for the following: Resolution will vary greatly between print and Web needs (you need four times as much resolution for high-end printing, and 10 times as much for output to the film recorder for re-creating negatives).
The following sections list the basic procedures for each image type using Photoshop.
GIF/Web graphic transparency:
Isolate the image area using a selection.
Remove the outer portions of the image.
Save it as a single layer in Indexed Color as a GIF with transparency.
Isolate the image area using selection, leaving the rest of the image intact.
Create a clipping path from the selection.
Save in EPS or TIF formats.
The net result for each of these (pending resolution) is nearly exactly the same when it comes to the use of the image in the media it is designed for.