For much of the year, distinguishing male and female American Goldfinches is easy (when the males show their brilliant yellow summer plumage, about March through September). Even in January and February many males have a few bright yellow feathers showing, but otherwise the gray-brown nonbreeding males can be hard to tell from females. There is very little difference between immatures and adults of each sex.
Males have really black wings with bright wingbars and feather edges, while the females have duller brownish-black wings with buffy or brownish-white wingbars and edges. This is pretty easy to judge, but it requires a bit of experience and judgment.
For a more objective and reliable difference, look at the underside of the tail, which is easily seen when birds are sitting on a feeder. Males have blackish tail feathers with well-defined white spots, females grayish feathers blending into dull white spots. Once you’ve confirmed the sex of a bird by the tail pattern, take a minute to look at the wings, the body plumage, bill and leg color, etc., and you’ll soon become an expert on goldfinch plumage variation.