Put your cut out onto a white background in Photo-Paint, then change the image mode to Palleted(8 bit) it will probably tell you it wants to combine the layers and that's ok. Next, go file/saveas and select gif, the gif export screen will now come up, under transparacy select image colour and with the pick tool click on the white area of your original image (it's asking what colour you want to be transparent)and then click ok.
The image will obviously still open within a rectangle in Paint (should be on a chequered background)so test it within your browser.
Graduating tones won't come out as well as jpeg's but if you want anything other than a rectangle there's not much else you can do.
If you have Fireworks you can play with the PNG formatt, good tones and trans, but not all browsers support it yet so gif is probably the go.
Hope that helps.