"We're Sgt. Pepper's Lonely Hearts Trope Band, we hope you will enjoy the show":
Aborted Arc: Apart from the Intro, the introduction of Ringo Starr in the persona of "Billy Shears" at the beginning of "With A Little Help From My Friends", and the Reprise, the album largely ignores the "concept" of the Sergeant and his band.
Album Filler: John only composed "Being for the Benefit of Mr. Kite!" (which basically transcribed a circus poster he had purchased at an auction) because he didn't have enough songs.
"Good Morning, Good Morning", inspired by a Kellogg's Corn Flakes commercial, was also this.
Alter Ego Acting: Paul devised a concept of the Beatles recording the album as if they were the fictional Sgt. Pepper's band, in an attempt to get the Beatles to rethink how they were going to work in the studio, and to escape the expectations of Beatlemania and start anew. The facade was somewhat abandoned of the songs took shape, aside from the title song, its reprise, and "With A Little Help From My Friends".
Book Ends: The title track and its reprise (with "A Day in the Life" being an encore)
Boring, but Practical: The line about "4.000 holes in Blackwood, Lancashire" was inspired by a newspaper article. Lennon was really flabbergasted that someone would spent his time to count the amount of holes he could find in a city. The boring activity did inspire one of the best songs on the album, though.
Concept Album: Besides the title tracks, it attempts to be like a concert. The concept was that The Beatles were pretending to be another band and doing the kind of songs that that band would do instead of just making an album as themselves. Although there is intense debate over whether it actually is a concept album proper, since apart from the two title songs and "With a Little Help from My Friends" the concept of 'the fictional band's concert' doesn't really develop much further; Lennon himself said that he wasn't working to this idea and that his songs, at least, could have easily gone on any other album.
Fleeting Demographic Rule: Variation. Before this album, The Beatles had putting out albums and touring almost unendingly, and they were feeling burned out. So, from this album onward, they decided to stop touring and take their time working on the album instead of working as hectically as they had been. However, since that meant the album took much longer to be completed and the band wasn't making any official public appearances during the process, the perceived decreased productivity of the band in the public's eye led critics to declare that the band was officially dead. These people were proven to be very wrong.
Genre Roulette: Music hall, jazz, rock and roll, classical music, Indian music...
Hidden Track: Actually the first ever example of this and thus the Trope Namer. After the final song, "A Day In The Life" a short little tapeloop can be heard: "I never could see it any other way", repeated until a Fade Out comes in.
The above is actually an attempt to reconstruct what happens on the vinyl album. By The Sixties most record players had an autoreturn mechanism on the stylus arm, but cheap or old models would not return when they went into the runoff region of the disc. Instead they'd circulate forever in a catch groove in the middle of the record, making an annoying distinctive click-kaclick-kaclick-kaclick noise as the record turned. One of the Beatles wondered if it was possible to press an actual short loop of sound into the catch groove. The result was if you played Sgt Pepper on a cheap 60s vintage record player, you'd hear it repeat that short weird message forever instead of the ka-click noise. Also, the moment of "silence" between the end of Day In The Life and the catch groove actually contains a noise pitched higher than humans could hear. The idea was it was supposed to set dogs off barking!
Ringo Starr preferred The White Album and Abbey Road, but for more personal reasons—he claimed that he felt out of the loop and "like a session drummer" during the recording of Pepper, a claim that becomes particularly hilarious when you remember what teeth-pulling frustration and fighting took place during the next albums' recording sessions. (Regardless, many people consider "A Day in the Life" to contain some of his best work.)
Harrison's ambivalence over the album was ultimately for similar reasons to Ringo — given the emphasis on lush, orchestral psychedelia on the album, there wasn't a lot of call for his skills as a lead guitarist, and since he only had one written song on the album he was nudged to the sidelines a bit as well.
Match Cut: The rooster crowing in "Good Morning, Good Morning" to the guitar of "Sgt. Pepper's Lonely Hearts Club Band (reprise)".
No Celebrities Were Harmed: All the celebrities on the album cover are photographic cardboard cut-outs or wax statues borrowed from Madame Tussauds. George Martin was so smart to ask every living celebrity permission to use their image. Only one person complained and asked a fee, actor Leo Gorcey, who was then painted off from the cover.
Non-Appearing Title: "A Day in the Life". The whole phrase "Being for the Benefit of Mr. Kite!" too.
Pun: "She's Leaving Home": The last word in the chant "we gave her anything money could buy" is repeated again after Paul sings the line "she's leaving home after leaving alone", sounding like the farewell word "bye bye".
Ripped from the Headlines: "She's Leaving Home" was based on a story Paul read in the paper about a girl who ran away from her home. "Being for the Benefit of Mr. Kite!" was based on a really old headline...specifically, a poster for one Pablo Fanque's fair that John saw in an antique shop. His verses in "A Day in the Life" are similarly based on various newspaper articles.
More specifically, the first verse is about the death of London socialite Tara Browne in a drug-fuelled car crash note at least for Lennon, who was friends with Browne; in the Wikipedia article, McCartney claims to have been thinking of a generic politician when he and Lennon co-wrote that verse. The second verse is an allusion to the movie How I Won The War (in which Lennon himself appears). The final verse is referencing a story about potholes in the roads of Blackburn.
Throw It In: Almost literally invoked in "Being for the Benefit of Mr. Kite!", when after several attempts to devise a suitably circus-sounding bridge by adding tape loops of calliope music, a frustrated George Martin told engineer Geoff Emerick to cut them up into small strips, throw them up in the air and splice them back together at random.
What Could Have Been: "Strawberry Fields Forever" and "Penny Lane" were recorded during the sessions for this album with the intention of them being part of it, but they ended up being released as non-album singles instead. Additionally, there was supposed to be a television special based on the album, but it was scrapped before completion. There was, however, a promotional video produced for "A Day in the Life" (as well as the aforementioned singles).
"Only A Northern Song" was briefly considered, too.
A sequel was also considered at one time, which would have featured "Baby, You're A Rich Man" (or at least its prototype, "One Of The Beautiful People")
The cover would originally also contain images of Adolf Hitler, Jesus Christ, Mahatma Gandhi and actor Leo Gorcey. The idea to put Hitler and Jesus there came from Lennon, who wanted to provoke. Eventually they were dropped because their images would certainly offend people, especially since Lennon had already hit controversy when his 1966 quote that "The Beatles are more popular than Jesus" was ripped out of context. Gandhi was dropped under pressure of EMI, who feared a backlash in India where Gandhi's status is practically divine. And Gorcey was airbrushed out because he was the only celebrity who wanted a fee for appearing on the cover.
World of Chaos: "Lucy in the Sky with Diamonds" even provides the quote for that page.
People appearing on the famous cover who have their separate TV Tropes article: