Sigal is yet another simple static gallery generator
You give it a directory of photos, it spits out HTML/CSS/JS for a nice static gallery you can point a browser at.
It has sensible defaults (read: ones that work nicely for me) while being easily customisable, it generates relative links so if you have an existing website you can still easily dump it in a sub-path, and it provides a whole bunch of small-but-really useful gallery features from thumbnail generation to EXIF tag extraction to bulk download generation and more.
Being a static gallery, hosting is trivial (I usually deploy to Amazon S3 but you could easily use GitHub Pages or many other static hosting options) and your exposure to security issues is negligible.
Thanks to Malcolm Scott for the recommendation!