Thanks to Wikimedia Common’s source of free photos, you can see the difference between the black-backed (left) and the green-backed (right) male Lesser Goldfinches below.
You wouldn’t like mistake a male Lesser for a male American Goldfinch, especially when both are in breeding plumage. (See a photo of a male and female American Goldfinch on the left; the male is left, the female, right.) But the females of the species look a lot alike, especially in their duller, non-breeding plumage, although the female American Goldfinch is a bit larger. (Female Lesser in photo on the right.) However, if they’re not sitting side by side, that’s not much help. One sure sign is the color of the beak. A female Lesser has a dark beak; a female American, a light beak.
You can hear a clip of the Lesser Goldfinch’s song and “tee-yeer” call here; scroll down a bit to the “typical voice” section.