This is the promo video with a live audience that they made for airing in Britain-- Michael Lindsay-Hogg of later Let It Be fame directed it. Then they brought in David Frost and a couple other TV hosts for the intros so they could each "introduce" the video on their shows. That's why David Frost is here messing around with John and George, anyway.
Since this is such an oft-written-about song, you probably already know that Paul wrote it for Julian, John's son, who was only 4 or 5 at the time and whose parents were divorcing in a fairly dramatic and public way. He apparently began writing it in the car on the way to see Julian and Cynthia, his mother, to see how they were doing, singing "Hey, Jules" to himself. He changed it to Jude just because he thought it was easier to sing.
Paul had a habit of writing melodies first and then trying them out on people with dummy lyrics (see "Scrambled Eggs"), and that's what he did with "Hey Jude." When he first played it for John, he apologized for the lyrics that made no sense, like "the movement you need is on your shoulder" and so forth, but John insisted that everything about it was perfect, and that he absolutely must keep everything as it was. And so he did. That's basically the story of "Hey Jude," except for a couple footnotes: that John for much of his life insisted that Paul had actually written the song for him (John) subconsciously (Paul has denied it), and that John also remained irritated up to his very last interview that "Revolution" was relegated to the B-side of this single. To which I say, get over it, because you're the Beatles and it's not like people were exactly IGNORING the songs that were B-sides. They're both tremendous songs. In fact, the "Hey Jude"/"Revolution" single might be the strongest single ever for them (unless you give it to "Penny Lane"/"Strawberry Fields Forever," which also totally makes sense). This was, by the way, the first single released on Apple, so it definitely got the label off with a bang.
"Hey Jude" works its magic within the realm of super-traditional chords-- it's in F, and there's nothing more adventurous here than a couple of g minor chords in the bridge or something. Otherwise it's basically I, IV, and V all the way. (It was easy enough for me to teach myself to play it on piano in high school, which is saying something, for I was never that great a pianist and certainly not great at playing by ear.) But there's a reason that composers use these simple harmonic structures so often: they are deeply, deeply satisfying. "Hey Jude" makes you feel so good largely because we just innately love these chords. Of course, they can also be handled in a boring way, but Paul avoids being boring in "Hey Jude," partly because his melody is so beautiful and singable, and partly because he's so typically smart about the details. Just a few of the things I love about the arrangement: the gentle entrance of Ringo's tambourine on the second verse, the downward scale the bass plays in the bridge (which becomes gut-wrenchingly awesome if you crank the bass way up), and John's backup vocal on the last couple verses, which starts lower than the melody and ends up higher. These are small things, sometimes just flourishes, but they provide exactly the right color, you know? They're just perfect.
I don't believe "Hey Jude" was ever conceived of as existing without the very long "na na na na" coda that makes it the longest Beatles single ever (the longest Beatles SONG ever-- not single-- is "I Want You [She's So Heavy]"-- hope these factoids prove useful in your next bar trivia game). The coda does introduce slightly edgier harmonic stuff, namely an E-flat major chord-- but I think the flat-major-seventh thing was something the Beatles did a lot, so it wasn't particularly edgy for them, and it sounds quite natural. Anyway, the coda is where the giant orchestra first comes in (again, what attention to detail! the orchestra would have sounded shit during the song proper, wouldn't it?). The voices singing along are also the members of the orchestra-- the Beatles asked them to sing along and clap after laying down their instrumental stuff. (The musicians got paid a double fee for it. Could you imagine getting to sing along to "Hey Jude" and getting PAID for it? Lucky skunks.) Paul whoops and wails it up with total earnestness as everyone na-na-nas, and it's freaking beautiful, isn't it? It's perfection. "Hey Jude" becomes some kind of anthem here, though for what exactly I'm not sure.
As he should, Paul remains proud of "Hey Jude" and plays it live pretty much all the time. I've seen him do it several times now live and on TV and stuff, and he's almost smug about how he can wrap the crowd around his little finger just by playing this thing. But that's okay. Rather than find one of those videos, here's one I like of some "Hey Jude" rehearsals. "Hey Jude," much as I love it, benefits from a little irreverence, don't you think? (I have no idea where this video originally aired, by the way, but I'm really glad it exists.)
"Hey Jude," released in the U.K. as a single c/w "Revolution," August 30, 1968; in the U.S. August 26, 1968.