To make the background of an image transparent, open the image for which you want to change the background. Note that the Layers palette has a single layer, "Background."
In the Layers palette, press Option/Alt and double-click the Background layer. This will change that layer into a "real" Photoshop layer that can deal with transparency. Note that the word "Background" in italics has been replaced by the non-italic "Layer 0." If you don't press Option/Alt when double-clicking, you'll get a dialog box that allows you to change the name of the layer when creating it.
Using any combination of selection tools (the Magic Wand/Lasso combination tends to work the best), select the background of the image.
If you want to add a semi-soft edge to the image, choose Select>Inverse to choose the image, choose Select>Feather and then enter a value such as 1 or 2 pixels in the Feather Selection dialog box. This will soften the edge of the image so that there won't be jaggies between the background and the image. After feathering, choose Select>Inverse again so that the background, and not the image, is selected.
Press the Delete (or Backspace) key on your keyboard. The background will be removed. When you save the image for placing in another Creative Suite application, be sure to save it as a .psd image so that the transparency is preserved.