"the covering that covers all peoples, and the veil that is spread over all nations" (25:7). What is it?

The verse doesn't say. Covering is lot, a noun the Hebrew Bible has only here, and covers is the verb it comes from, which Brown-Driver-Briggs gives as "wrap closely, tightly, enwrap, envelope" (tap covering). Veil is massekah, which Brown-Driver-Briggs gives in this sense as "woven stuff, web, esp. as covering". What the verse does say is who takes it away, where, and from whom: "He will destroy in this mountain" (25:7), from over "all peoples" and "all nations" (25:7). The next words, with the same Hebrew verb, are "He has swallowed up death forever!" (25:8), and the tears are wiped "from off all faces" (25:8). The verses set these side by side and say no more about the covering, and these notes don't either.