Depends on what you call essential. AFTP would have the most hits, but R.E.M.'s best and most influential work was done in the '80s. Hmm… why don’t we make this easy and have a couple R.E.M. ones. They have been one of the defining modern rock bands.
I would suggest the following, some of which have already been suggested. I have included quite a lot of records from the '00s, but that’s because I think great rock continues to be made to the present day. i have thought my selections through and I consider my recent selections to be just as worthy of inclusion as the older ones.
The Clash - London Calling
Obvious place to start, and even though it was released in the UK in '79, it’s US release was 1980, so it squeezes in. I just noticed there’s a punk thread planned, so perhaps this should be held off until that, but the record is diverse enough to be considered under the modern rock banner.
U2 - The Joshua Tree, Achtung Baby
U2’s best records and most representative. War would be a good third choice, particularly because it shows the groups early more post-punk sound, but it’s more an essential U2 record than an essential modern rock record.
R.E.M. - Lifes Rich Pageant, Out of Time and Automatic For The People
They really do deserve three albums here. Most modern rock is basically permutations of The Pixies, R.E.M., and more recently, Radiohead. LRP may not be as good as Murmur but its the best introduction to their '80s sound, which really made them famous. It also has the better singles, and is a pretty good representation of the band - “Begin The Begin” shows their rocking anthemic side, “Cuyahoga,” their jangliness, “Fall On Me,” their ability to write unforgettable singles. Out of Time has their biggest hits, and some of the biggest hits of the '90s - “Losing My Religion” and “Shiny Happy People” as well as classic like “Near Wild Heaven.” Automatic For The People was the completion of their transition into the mainstream, and is a great album, all six singles are essential and is one of the strongest, most musically and thematically consistent and cohesive records of the period.
The Cure - Disintegration
I think this is generally agreed to be the classic. It’s tricky though, because The Cure are also a great singles band.
The Smiths - The Queen Is Dead
The Replacements - Pleased To Meet Me
I’m putting this in tentatively. Love the Replacements, but I get their records confused a bit. They certainly need to be on this list.
Sonic Youth - Daydream Nation, Goo
Nirvana - Nevermind, In Utero
I would argue that both records need to be included, to show both the catchy, hooky side of the band and the reactionary, noisy side.
Pixies - Surfer Rosa, Doolittle
Wrote the blueprint for indie rock.
My Bloody Valentine - Loveless
Pearl Jam - Ten, Vitalogy
The Smashing Pumpkins - Siamese Dream, Mellon Collie and the Infinite Sadness
The greatness of SD seems to be accepted, but some people are less fond of MCatIS. It could be argued that at a double it is too bloated, but even if you do take issue with that, the quality of the songs is undoubtable - Zero, Bullet with Butterfly Wings, 1979, Stumbleine, Tonight Tonight
Beck - Odelay
Pavement - Crooked Rain, Crooked Rain; Slanted and Enchanted
Radiohead - The Bends, OK Computer, Kid A
I think all three of these must be included. The Bends showed their greatness as a pop group, OK Computer remains one of the greatest albums of all time, completely flawless (yes, even Fitter Happier is perfect) and Kid A is the prickly, left-field equal to Ok Computer. I must emphasize this point - as far as I am concerned, there is no doubt that Kid A is as good as OK Computer, if not better. Not only does it have great songs - Idioteque, Everything In Its Right Place, The National Anthem, as an album it is an astounding experience, 40 minutes where everything is in its right place. So what if it doesn’t have guitars?
Mogwai - Young Team
Neutral Milk Hotel - In The Aeroplane Over The Sea
The Flaming Lips - Yoshimi Battles The Pink Robots
I like it better than the Soft Bulletin, although that record could easily substitute for this one.
Wilco - Summer Teeth, Yankee Hotel Foxtrot
Modest Mouse - The Moon And Antarctica
At The Drive-In - Relationship Of Command
Every time I return to this record it amazes me again with its power. How sad that they had to become the disappointing Sparta and the wretched Mars Volta.
Interpol - Turn On The Bright Lights
Almost gets passed over because it just perfectly works a sound that has been done before, but the quality of the songs make this an equal to the other records on this list.
Red Hot Chili Peppers - Blood Sugar Sex Majick
The White Stripes - White Blood Cells
The Strokes - Is This It
**The Arcade Fire - Funeral
Death Cab For Cutie - We Have The Plans And We’re Voting Yes
Built To Spill - There’s Nothing Wrong With Love**
There’s nothing wrong with Built To Spill in their more adventurous, guitar-hero mode, but I prefer their indie-pop simplicity. Substitute with Perfect From Now On if you wish.
The Shins - Chutes Too Narrow
They’ll change your life. 
**Wolf Parade - Apologies To The Queen Mary
Bright Eyes - Lifted, Or The Story Is In The Soil, Keep Your Ear To The Ground
Belle and Sebastian - If You’re Feeling Sinister**
Feel free to substitute Tigermilk.
Weezer - Blue Album, Pinkerton
Both are vital.
**Billy Bragg - Talking to the Taxman About Poetry
Jeff Buckley - Grace
Blur - Parklife
Yeah Yeah Yeahs - Fever To Tell**