My personal favorites:
Great Teacher Onizuka - First manga I ever read and it had me in freakin tears. Basically, an ex-biker gang leader decides to become a high school teacher and is put in charge of the most unruly class at a private school. Hilarity ensues with a few good bouts of drama thrown in for good measure.
Angel Densetsu - Created by the guy who did Claymore. It's about a high school boy who has the face of a demon/devil/super evil dude and is mistaken at his new school for a gang leader, murderer, hard core druggie and etc. It has moments of action, some drama, lots of comedy, and a pretty solid cast of female and male characters. A personal favorite of mine. (I think, I shall start rereading this one now.)
One Piece - I don't think I have to explain the story about this one since it's so friggen popular so I'll explain why I love it so. First, is the artwork. It is so distinct and yet diverse. I love how no two characters look the same and you get an enormous plethora of body types, looks, and etc. (Though the change in look of the two female leads has been a bit of disappointment imho.) It also lends itself incredibly well to the series' humor. Next is the world building, Oda has done a great job building a very believable world and one that I have seriously enjoyed exploring from the very beginning. To me, it's one of One Piece's strengths. And last is the storytelling... I cried over a friggen boat. I don't think much more needs to be said lol. Also, the anime is a TON of fun! I have started collecting it season by season. I know it's a helluva long series but honestly, I would totally recommend it. (Just skip the Davey Back Jones arc, it's the shittiest of the bunch. Instead watch the G-8 arc as to me, despite being a filler arc in the anime, it's a great addition to the One Piece universe and totally worth watching.)
Berserk - Ooooh how awesome this series is. The artwork is phenomenal as is the action, the total badassery of the main character Guts, and all of the super craziness that has happened since the sacrifice. Also, the original anime series trolls you so hard XD, it's great watching people's reaction to the series for the first time haha. Good stuff. Lastly, some don't like the movies but I must say that they are still pretty damned good and worth the watch. The first movie is available on Netflix right now for the those curious. Both the movies and the show miss a TON of stuff but they both do a great job of complimenting the manga.
School Rumble - One of the funniest mangas I have ever read. It also includes of my favorite female characters, Eri Sawachika. It had a terrible ending but I had read that the publisher forced the author to rush it so it had a crap ending. Thankfully, he made a much more proper ending in School Rumble Z and ties up a lot of loose ends. Overall, a great read.
Aqua/Aria - Nez already mentioned this one but I just have to relist. A beautiful series to read and just look at.
Yokohama Kaidashi Kikou - YKK for short. Another slice of life masterpiece. Both it and Aria to me, are the best slice of life stories I've read. It takes place in post apocalyptic Japan well after the apocalypse settled down and people are attempting to etch out a living. It follows a female android as she runs a small cafe in a very rural area as she goes about her daily life. A very good yet, melancholic read. No crazy drama, action, romance, or anything of the sort. Sometimes there are entire pages where nothing is said.
A Bride's Story - By the author of Emma: A Victorian Romance, this follows people living in Central Asia in what I am guessing is the mid-late 1800's. It has gorgeous artwork and an engaging story. Very well worth the read. I would also recommend Emma as well as it's a very good read and it has a much more satisfying ending than the anime despite the anime having a pretty good ending as well. Also, some might find it difficult to accept a prearranged wedding between a young 20-something year old woman and a 13ish year old boy but when read in the context of the setting and time period, it makes a lot of sense and also for a very interesting read.
Pumpkin Scissors - A pseudo-European setting that follows a military branch, known as Pumpkin Scissors, that focuses on post-war reconstruction following a devastating war. The main character is the hulking man who is actually very kind and a bit clumsy however he has a very dark past in relation to the war that his comrades in the Pumpkin Scissors are slowly discovering. However, there is some underlying conspiracy going on and the main cast of characters keep bumping up against it. Sadly, the anime doesn't get very far but the manga is still on going, but sadly it seems it's stopped being translated.
Gunka no Baltzar - Another pseudo-European setting this time in an era much more similar to World War I. It follows an officer of a country modeled heavily off of Germany/Prussia who is tasked with building relations with a neighboring country. He is of course, ordered to be an attache/adviser at the smaller country's sole military academy. So far it's been pretty good. I've enjoyed it quite a bit. It has had good artwork, lots of action, drama, comedy (if you can stand its comedy), and politics, and drama, and etc.
Baggata Way - A sports manga that follows an all-girls lacrosse team as they build friendships, discover the sport, compete, and have fun. Of course the main girl is a bit of a natural which I don't mind at all. Lots of comedy and exposition on the sport as well. One of the only sports mangas I read. It's available here as well but sadly, translations are a bit slow to very glacial to very non-existent.
Those are, I would say, my favorite mangas outside of the ones found here. (Check out my classics list if you want to see my favorite yuri stories.) There are plenty more that I would list but this top out as my all-time favorites. Below are a few other things I am reading that are in different formats but worth it imho:
Girl's of the Wilds - Webtoon - A martial arts action Korean webtoon that follows a boy who transfers into a newly minted coed school that was originally a totally girls only school. However, unbeknownst to the man guy, it's heavily oriented towards martial arts and all of the girls there are top athletes basically. It's been going on for a while and it definitely has its slow points, low points, and dull points but! It also has great comedy, drama, and action plus a solid female cast. I enjoy it immensely.
The Irregular at Magic High School - Light Novel - After watching the anime I chased down the light novels because I had heard that despite being a pretty decent anime, it missed a TON of information and scenes in the light novels. And, it was true! A LOT of very good bits of information and some great scenes were either left out entirely or just portrayed totally wrong. It really like the setting but the bro-siscon/incest thingy is kinda really weird sometimes -_-...
My Youth Romantic Comedy is Wrong as I Expected - Light Novel - Of course, only discovered it after the anime but I really enjoy the read. Hachiman is quite the main character and I can really relate to some of his dialogue and interactions with other characters. Some pretty interesting things going on. I would personally recommend it over the anime because the anime skips a LOT of moments. Unlike Irregular, the moments missed in the anime are actually, imho, moments that should have never been skipped because they add a lot to the series. So, watch the anime as a companion to the LN.