Cube
Revenge of the Sith: People can't seem to appreciate that it's a masterpiece.
Unbreakable
Donnie Darko

Some really good recommendations in here, one of the previous posts of Trainspotting made me think of Shaun of the Dead actually, because both films are very popular in the UK which is where I'm from but may not be well known around the world.

Shaun of the dead (2004)is a Zom-Rom-Com, quite a standard affair really, however it is a great comedy and a must watch imo

https://www.youtube.com/watch?v=LIfcaZ4pC-4

The Conversation (1974)

Next, a film directed by the very well acclaimed Francis Ford Coppola, The Conversation.This film came out between the release of The Godfather part I and The Godfather part II, and was overshadowed by them completely (admittedly, they are the better films).
Plot, copied from IMDB - A paranoid, secretive surveillance expert has a crisis of conscience when he suspects that a couple he is spying on will be murdered.

I'm posting a trailer for The Conversation below but I wholeheartedly recommend not watching it, it really is the type of film that you need to just sit down and watch without knowing too much about it.

https://www.youtube.com/watch?v=lwfI5YwvcrI

The Thin Red Line (1998)

This is a film that is based in the Pacific during WWII. Directed by Terrence Malick, the film takes on a much more philosophical take on war, what it is to be alive and why does nature compete to survive rather than live in harmony? It's safe to say that Saving Private Ryan stole this films thunder and to be fair, maybe this film isn't for everyone, but I love it and think it is the best war film that I've seen, even better than Apocalypse Now (an unpopular opinion, no doubt).

https://www.youtube.com/watch?v=mKl5_OxKBn8

Brazil (1985)

This one is directed by Terry Gilliam, based in a dystopian future that is spookily similar to ours now, but in a funny observant kind of way. The plot is about a man who is arrested and took away by the police but they get the wrong guy due to an administrative error. Bring in our protagonist, Sam who makes it his business to get this wrong righted while also searching for the woman of his dreams. I can't really explain the full scope of the movie because there's so much going on, but It's a funny film that really worth the watch and I recommend this strongly.

https://www.youtube.com/watch?v=ZKPFC8DA9_8

And finally, Embrace of the Serpent (2016)

Ok so this one has what seems like a dull story. It is based in the Amazonian region across two timelines where one central character, Karamakate helps two western explorers to find a somewhat mythical plant. The film is more about the journey that these characters take together and the culture differences that they have like differences in religious beliefs - really thoroughly recommend this one, but can't explain to you well enough, so beneath is a review by Britains most well respected critic, Mark Kermode, and beneath that, a trailer

https://www.youtube.com/watch?v=kQnwo3vGR_k

https://www.youtube.com/watch?v=uOa9pjl37Lo

Well, not really the best of all time but I like it (Blood in, Blood out)

The film and tv series adaptations are terrible. They don't do the books justice. So much is lost without a detailed inner monologue and thorough analysis of minutia in critical moments. 

Primer is great, but I prefer the director's follow up movie Upstream Color. I'd say its as hard to wrap your head around as Primer but in a very different way. Its one of my favorite movies and most certainly underappreciated.

https://www.youtube.com/watch?v=76pCrk7P_5o

Octalus

I'll definitely check this out

Brotherhood of War
The best war movie I have seen, I'm lucky to have randomly rented this one.
http://m.imdb.com/title/tt0386064/

Splice
Great gene-splicing sci-fi that I begrudgingly watched at my partner's request, but ended up loving.
http://m.imdb.com/title/tt1017460/

OldBoy
Not the shitty remake. This probably doesn't count as under appreciated, but it's hard for me to tell as no-one I know personally has even heard of it. Do yourself a favour and watch this madness. Also, the octopus is real.
http://m.imdb.com/title/tt0364569/

Catfish
An excellent documentary/mystery thriller. The show it spawned is average, but the movie is great.
http://m.imdb.com/title/tt1584016/?ref=m_nv_sr_2

Orgazmo
Probably Matt Stone and Trey Parker's best work. This is comedy gold.
http://m.imdb.com/title/tt0124819/

I'm glad you brought this up. Studied this film in college as a case study in english class. Great fim.

lol the problem with "underappreciated" movies is generally they're eventually discovered, become appreciated, and shortly thereafter are "overappreciated".

A Christmas Story, for instance, went from an under-the-radar solid Christmas film to a grossly overrated one. The internet is also really helping shine some light on those films that were missed.