Characters - I loved how deep they were compared to any other game I've played. And it went even further with subsequent games. Only in ME2 I understood that I grew attached to these characters. Little hints on previous events, cheesy lines, sense of trust and loyalty - I never experienced anything like that in a video game. Then came ME3 and it was my most emotional journey of all my gaming life. Sadness from Mordin's sacrifice, mix of sadness and laughter at Grunt's cameo, seeing Jack at Grissom Academy along with Kahlee Sanders who I knew from the books, Garrus's "Are you kidding, I'm right behind you" line on Menae, becoming furious on Kai Leng after Thane's death, Thessia and that blasted e-mail from asari high command (an awesome little touch, thanks Bioware!) and putting all that fury into that very satisfying renegade interrupt in the end...
Universe - I loved how the universe was believable, probably because it was based on real world science, the history described in the codex providing you background on races, technology and galactic society.
Story - Unexpected decisions, like Rachni queen dilemma, Normandy being grounded, revelation of Citadel trap, seeing Normandy destroyed and rebuilt, finding out the truth about Collectors, seeing and talking to a Prothean, brokering a peace between quarians and geth...
Sense of exploration in ME1 and ME2 - even more in ME2, because there were no more reused environments and each world had something different, like that one with a derelict ship, one with a quarian from Idenna (a ship from Mass Effect: Ascension). There were also great hub worlds - Illium and Omega!
Music - the series have the best soundtracks I've ever heard in video games. Even after all these years I get goosebumps during the Suicide Mission 
Voice acting deserves a separate spot for how great it is. I can't even mention a character with sub-par voice acting!
Overall professionalism - very small amount of bugs, timely patches, nothing game-breaking (unusual experience after Skyrim), improvement over series, DLCs that actually serve not as money-grabs but as testing grounds for new games.