A graphic library: ImageMagick is recommended for its performances and image quality but GD, often bundled with PHP, can also do the job.
an FTP client software will be required to upload the files (netinstall or full package): Piwigo team recommends FileZilla as FTP client software, because it is free as Piwigo is, and compatible with Windows, Mac and Linux.
Enough disk space for your pictures: in addition to the photos you upload, Piwigo will store the "multiple sizes" in a cache directory of your server.
exiftool is required for plugin Write Metadata or any other plugin dealing with EXIF/IPTC metadata
ffmpeg is required for plugin VideoJS to create video poster
pdftoppm will be recommended to generate poster from PDF files (instead of ImageMagick)