Yes, you are right.

A new problem now: the image is displaying something, at least, but it's the wrong thing. It draws a diagonal line with a white background (there's supposed to be no background). The line goes from the left side to the top side (although I'm using axes where 0,0 is in the top left corner). Any ideas?

oh, oh, oh, wait a second. I got my glTexCoord2i()s mixed up. Sorry.

Glad you got the glTexCoord*() business sorted out. When you say that the image should have no background? Do you mean you should just see the line, and then anything behind the image? If this is so, then you need to make sure that the image uses an Alpha, supported in such image formats as PNG, GIF and TGA. You also need to remember that you'll need to specify GL_RGBA as the format for the texture.