I'm really surprised that no one here recommended
Bokura Ga Ita, that one made my cry (from joy and sadness) quite a few times
Here are some more that I recommend:
Air TV - probably not for everyone, but the love IS there, and it was quite sweet
Kare Kano - one of the classics (but I must admit the ending was a bit disappointing, but there's always the manga
)
Midori no Hibi - a really funny one, but the romance is really good
NANA - now that's what I call real human emotions, the love stories in this anime are so realistic that it hurts
Paradise Kiss - from the creator of Nana, so yes, it is good
Saikano - another classic, with a beautiful ending
Honey and Clover - really realistic, probably not everyone will love it, but human relationships, are really good in this one