Photobucket will do nicely, to coin a phrase.
I don't use photobucket, but I understand that when you upload a picture to your account it gives you the URL (web address) for that picture. This will be a line of code in the format you're used to seeing in the address bar of your browser for any web page. But instead of ending in .htm it will end .jpg or .gif. So look for the line beginning http:// and ending with the picture type. That's your image address.
To include an image in a text box (as opposed to using the built-in one per module widget) you have to link to your picture using an html image tag.
In it's simple form this has the format shown here:
Code:
<img src="http://www.restofyourpictureaddress.jpg">
In an ideal world you should add other stuff, but that will work as is and keeps life simple for the beginner.
Just replace the obvious bit with your pictureaddress from photobucket.
If you have difficulties with photobucket, you'll have to ask somebody else, I'm afraid, I don't have an account with them.
Good luck!