OpenSSL
Dominant TLS + crypto library. Open source since 1998. Powers practically all HTTPS on the internet (nginx, Apache, Postfix, OpenSSH, every language's bindings).
Famously complex codebase with classic security incidents: Heartbleed (2014, CVE-2014-0160), CCS Injection, Logjam. Spawned alternatives: BoringSSL (Google fork, internal), LibreSSL (OpenBSD fork after Heartbleed, code cleanup), wolfSSL (embedded), AWS s2n (minimal). v3.0 (2021) modularized provider architecture, FIPS validation. CLI: openssl s_client, openssl x509, openssl genrsa — the Swiss army knife for cert debugging.