Broken freestanding single handle

Severe layman here. This free-standing faucet handle in my sink has been leaking for a couple days, and whole thing broke off this evening.

I'm not scared to crawl under the cabinet and turn a wrench to replace this small thing, but I don't know what words to search to even find what I need to buy.

Can someone help guide me with some better DIY search terms to fix my problem?

broken handle