This is a quick break down of the marks to look for that seperate male House Finches from the similar in appearance male Purple and Cassin's Finches. House Finches are common backyard visitors and abundant throughout most of their range so familiarizing yourself with its variations and key marks can help answer the age old question of; House, Purple, or Cassin's? I also advise looking at range maps to see which is likely in your area as well as when.
1. Bill Shape - The ridge of the upper mandible (culmen) is notably curved on House Finches, giving the bill a slightly thicker stubby appearance. On Purples and Cassin's it is straight giving the bill a more conical look.
2. Streaking/Color - The color of a male House Finch is extremely variable in both coverage and hue, although typically average a lighter pinker red than Purple. Some males show very little red limited to the head and the very upper breast, while some more heavily colored males show a deep red that extends all the way down onto the belly; and in extreme cases, the undertail coverts. Unlike the males of the other two species, male House Finches typically show a good amount of dark brownish heavy, indistinct streaking on the belly and flanks. Areas of intense red can often wash out streaking (as we see the diffused streaking hidden on the breast of this male) so more brightly colored males may show less streaking on the belly, but typically show moderate streaking in the flanks, which is a good mark to look for.
3. Auriculars - The auriculars (cheek) are typically grey and notably paler in contrast to the surrounding areas of red. They may occasionally have a pinkish/faded brownish wash but never a deep red or rich purple brown.
4. Back and Wing Coverts - The red coloration is often limited to the nape and some spilling onto the mantle (center of the back) and filling it on more colorful individuals, but rarely onto the scapulars (shoulders) which are typically grey/brown in coloration, like our male above. The red color never reaches down into the wing coverts which show a buffy or white edging (making up the visible wingbars) with a grayish brown/black center, never with a notable purple/red tint or edging.
5. Undertail Coverts - Streaked undertail coverts in all plumages. Useful for separation from Eastern populations of Purple Finch which typically (though not always) show clean white, however; Cassin's and the Pacific population of Purple Finch also show streaking in the undertail coverts so this mark is most helpful in the East.
6. Tail - House finches are the longest tailed of the three with very broad rectrices (tail feathers) that taper bluntly at the end creating a shallow, to no, fork at the end depending on how the tail is held. Compare this to the sharply tapped and slightly graduated tails of the other 2 that create a nice deep fork with sharper points (perhaps most noticeable from behind). This is a bit of a subjective mark so be sure you are very familiar with the variation before applying it as anything but a tentative supporting mark.
To wrap things up House Finches are no exception to variation, but if you combine multiple marks and keep an open mind to the variation limits, as you adjust them to learn what most fall under these brightly colored finches can be identified with confidence. If you are ever in doubt, never be afraid to ask fellow experienced birders for their opinion. Ask for multiple opinions until you can begin to feel confident in the seperating of species on your own. Don't feel discouraged when you get them wrong either, even experienced birders are fooled by a Haemorhous genus finch time to time (and got fooled more then once on the journey to seperating them with confidence).