Is H264 an actual file format? Is there an actual H.264 file or is it just the codec that renders the video into an .mp4 or .mov file? 

For an application, I have to hand in a portfolio with videos. Their website says that, "Videos have to be H264 format and pictures JPEG." Now, JPEG is a file format as well as a compression codec. But I've never heard of H.264 as a file format before. Is it possible to render video into an "H.264 file" or should I encode my videos as H.264 into an MP4 container?
I'd bet cash money they don't know what they're talking about when they say h264 format. There is a raw h.264 bitstream format, with the file suffix .264, but it's pretty obscure (I can't duckduckgo any actual documentation on what it's for, possibly when dealing with live streams from cameras). 

Give them an mp4 encoded with h.264 codec - it's a format you can be pretty certain anyone with software that's not completely obsolete can play. Conversely a .264 stream is fairly well guaranteed to be unplayable in all but rather specialist settings.

But if you have any doubts you could ask them if h.264 encoded mp4 is what they mean.