These filters, from what I can gather, are effectively powered by the popular online image editor Aviary. It works in basically the same way as Instagram, except it's integrated right into the Twitter app. The filters themselves are different, of course. There are standard ones like black and white (Inkwell in Instagram) and warm (similar to "Toaster"), but they've also got vignette, cool, vintage, cinematic, happy and cool. Also like Instagram, the built-in image editor and filter adder lets you do basics like cropping and zooming.
The official Twitter app is still a free download from the Google Play Store. The update to the iOS version for iPad and iPhone should be along shortly too.