1. The OSTs, BGM, and in general the music of a series. If a series has music that I'd go back and listen to, I'll definitely remember it. A couple examples for me would be Guilty Crown and Shingeki no Kyojin.
2. Epic scenes or a series where I'm constantly on the edge of my seat waiting for what's going to happen next. The most recent anime I can think of would be Shingeki no Kyojin. Man, that series was awesome
3. Last but not least, and probably the biggest thing to me, is if I can take something positive (generally morals or values) from a series and try to apply it to my life. Naruto overall has been influential on me, especially the whole "never give up" message. After (and still) following it for so long, I found that I've become much more understanding and tolerant of others (though I'm sure it wasn't the only factor in my change). Basically, a series with a number of emotional and heartfelt scenes stands out to me, and after finishing it I get that empty feeling for a couple days.