My first suggestion is a rather generic choice, The Magic Portal by Lindsay Fleay.
While it's not the first brickfilm ever created, I think it has the most influence of any of the early ones. An example of this is the little head jump commonly used to indicate fear - as far as I can tell, this was first done by Fleay at around the 10:55 mark.

That's The Way It Goes by Walter Benson.
Also the entire JOE BRICKMOND series.

Henri and Edmond - Copyright has to be included, it's one of the most impressive brickfilms ever made. HD, but no english subtitles

Youtube Channel

The accidental exclusion of this brickfilm was keeping me up. Just had to come back on here and post it. Not only does it have one of the greatest dramatic stories I've even seen, but that fighting animation is definitely something to learn from, even if it is a bit more "fluid" than I'd like most punches/kicks to be.

Really, anything from Robinson Wood could qualify for this list.

I like seeing new threads like this be made, to see what the current perspective of things is. I've seen multiple similar threads over the years, so I thought I would make a list of the films that get mentioned the most often, plus a very small number added for their historical importance. I have stuck to pre-2010s only, as that is my area of expertise; I would really like to hear what other peoples' opinions of the essential viewing of the 2010s is. This list goes in chronological order, and it is not meant to be a definitive list of everything that ought to be mentioned from the era, as I have tried to be as strict as absolutely possible:

Journey to the Moon by Lars and Henrik Hassing (1973)
The Magic Portal by Lindsay Fleay (1989)
Twelve Bucks by Rob Weychert (1998)
ONE: A Space Odyssey by Spite Your Face (2001)
Monty Python and the Holy Grail in LEGO by Spite Your Face (2001)
The Gauntlet by Jay Silver (2001)
The Han Solo Affair by Spite Your Face (2002)
Fell in Love with a Girl by Michel Gondry (2002)
Rise of the Empire by Jay Silver (2002)
Good Company by Nick Maniatis (2002)
Out of Time by Chris Salt (2003)
The Letter by James Maduzia (2003)
Zero Gravity Rebellion by James Maduzia (2003)
Taco Trouble by Andy Boyer and Dave Lennie (2003)
The Citizen of the Year by Watson Films (2004)
The Peril of Doc Ock by Spite Your Face (2004)
Star Wars: The Great Disturbance by Leftfield Studios (2004)
Aladibababad by Leftfield Studios (2004)
Totentanz by Steffen Troeger and Andreas Mooslechner (2004)
Cognizance by Michael J. Green (2005)
Dr. Jobs and His Scary Laboratory by Fumitaka Anzai (2005)
Frankenstein by Robinson Wood (2005)
Cleaning Time: The Janitorial Contingency coordinated by Doug James and Joshua Leasure and directed by Tom Dean (2005)
Robota by Marc Beurteaux (2005)
Grace by Robinson Wood (2006)
Faithless by Matthew Gray (2006)
America: Outlawed by Doug Vandegrift (2007)
Little Guys! by David Pagano (2007)
Hastings by Nikolas Jaeger (2007)
Unrenewable by Philip Heinrich (2008)
Pirates! by Doug Vandegrift (2008)
802.701 by Steffen Troeger and Sandra Abele (2008)
Gaston's Odyssey by France & Alex (2008)
Alles ist die Noppe by nichtgedreht (2009)
The River by Nikolas Jaeger (2009)

I will also mention 1xY, The White Ninja, Great Inventors: Part 2, The Hand of Fate, Fallen: Part One, Special Deliverance, Triumph of the Empire, The Artist, Driven, The New Neighbour, Bill Carney's Body, Playback, and Days of Our Pizza. I really could go on, but instead I'll recommend taking a look through the Brickfilms by year category on the wiki if you want to see more of the older notable films. Obviously the wiki doesn't even nearly have every film page it should, but I have given particular attention to the pre-YouTube era due to the information (and often times, the films) being much harder to find than just all being on the one YouTube page. I think 2001, 2002 and 2003 are pretty close to having everything pageworthy.

As for the 2010s, of course Copyright has already been mentioned (I believe it is the greatest brickfilm). HOO also mentioned That's the Way it Goes!, which I agree with. Since this is an essential viewing thread rather than a definitive classics thread, I will give my own suggestions of some more films going up to this year:

The Force Unleashed by Jordan Johnson (2010)
The Pilgrim's Son by Jason Boyle (2010)
The Profession by Zach Macias (2010)
Egyptian Holiday by Spencer Katz (2011)
Viva la Vida by Dylan Woodley (2011)
The Duel by David Clark (2012)
The Call of Farqunglu by Kris and Kurtis Theorin (2012)
Bestia by France & Alex (2013)
The Front by Gregory and Garry Moore (2013)
Coffee & Carnage by Jonny Paquette (2014)
Pokeballin' 2 by Dylan Woodley (2014)
Plastic Love by Maxime Marion (2014)
A Fixed System by Aaron Fisher (2015)
Beyond the Eleventh Dimension by Chris Boyer (2015)
Minecraft: Creepers by Thomas Evans (2015)
The Revenge Song by Enrique Fuster (2016)
Sheep by Maxime Marion and Kloou (2017)

I'm hoping to see more 2010s stuff in this thread, as I feel there is so much more that needs to be mentioned and I don't know all I should, and I've seen less discussions coming to a general consensus on what the classics are than I have for the pre-2010s.

I second The Profession, The Front, and Beyond the Eleventh Dimension as some post-2010 brickfilm masterpieces.

Also, And Men Loved Darkness is a really great noir. You don't see those too often, you know? And, they're rarely done as well as Pritchard's film is.

Penta has provided an excellent list–essential brickfilm viewing for sure.

In terms of 2010s brickfilms, and in the spirit of Halloween, my favorite is probably
That Groovy Kind o' Love by Jason Boyle (2010)

I'd also recommend:
Stranger Than Fishin' by Zach Macias (2010)
Coffee With the General by Ross Pearson (2010)
Odd Jobs (minimum wage = minimum effort) by Jon Clark (2010)
Printed Circuit - Brick It by Chris Salt (2010)
Major Malfunction by Kyle Cullum (2012)
That's the Way it Goes by Walter Benson (2014)
Paint. by Jon Rolph (2015)
Paragraph by Steffen Troeger (2015)
Sola Luna by Rioforce (2015)

Although I'm not the biggest fan of this brickfilm, I'd recommend Robota over Sola Luna as fairly recent contest winners are kinda hard to tell the longevity of.

Fame, Infamy, and Glory is a bit different; spawning 2 really well established brickfilms. Beast, already on this list, and Grace - presently not on this list, but a definitely good candidate. Then again, Fame, Infamy, and Glory IS over 10 years old, now...

Brickfilms only about 2 years old are hard to tell if they'll still be well received by the time the next batch of brickfilmers joins the site/the craft. Although, I'm not totally against suggesting or even including some post 2015 brickfilms on here. I'm just not sure how well they'll hold up in the long run, you know?

I second the White Ninja, along with pretty much all of the others previously mentioned (except perhaps some in the long lists that have been posted). I had never actually seen Grace before, and it is really good!

Conveniently (or maybe not so , as there are hundreds) I have been working on a playlist of brickfilms that I recommend people watch. You can find it in the playlists section of my channel. Obviously I am definitely not suggesting that they all go on there, but if you want you can have a look and see if there are any that you think would fit the list - but I don't mind if you don't want to, as there are A LOT on that playlist

Everyone else is free to take a look, it's not finished, but there already some on the list made by fellow BiM users, so some of you can try and find any of yours on the list (I have also grouped together all brickfilms made by one person together).