An iPod classic can generally play video files up to 640 by 480 pixels. The iPod's screen is actually 320x240 pixels, but it will play 640x480 videos (that are in a compatible format).
However, "HD" (high def) videos are at a higher resolution than 640x480. Apple often provides both the HD and SD (standard def) versions of the video, when you download it. The SD version should work with iPod classic. If a video is in HD only, it won't play on iPod so you can't sync it to iPod.
There is a command in iTunes to convert the video. Select video on the iTunes library list. Go up to Menu Bar. Under Advanced, select Create iPod or iPhone Version. After converting completes (which can take some time), try syncing that version to your iPod. NOTE: Some videos in iTunes may not allow conversion due to copy protection (DRM).