The JPG format doesn't support transparencies, so you will have to use GIF, which will limit your selection of colors. Don't worry, though, it won't be drastic.

1. Open an image. "File | New" with any dimensions.

2. Select the oval selection tool by holding down the mouse and move over to the dotted oval symbol:
   

3. Make a selection by holding down the mouse button on the image and drag it:
   

4. "Edit | Copy" then "File | New" Make the background Transparent, instead of White or Background Color. The dimensions will be those of the oval you just copied. "Edit | Paste" into the new image.

5. "File | Export | GIF89a Export" A dialog box will appear:

6. Double-click on the grey box and choose a color that is close to most of the colors in your background. Choose off-white for mostly white backgrounds, and dark grey for mostly black backgrounds. Experiment with different colors and see which makes the edge seem the least jagged.

7. Click "OK". You might experiment with the "Preview" and different amounts of color. Decreasing the colors will make the graphic load faster, but sometimes it might make them look work. Previewing it will help so you don't have to keep going through the steps.

8. Another dialog box will appear, so just go through the folders and save it just like any other file.

Notice that I didn't use feathering on the oval selection. Using feathering with transparent GIFs can make the edge around your image look strange. There is no partial transparency. It's either transparent or it's solid. So if you have an image with a lot of feathering it will just make the partial pixels solid. Sometimes it can still be hidden, and Photoshop 5.5 deals with this slightly better, but this is the limit of transparent GIFs.

Next time, at long last, I'll be teaching you the magical shortcuts in making seamless frames.