Both "ai （愛）" and "koi （恋）" can be roughly translated as "love" in English. However, they have a slightly different nuance in Japanese. 

Words containing "ai"

aidokusho 愛読書 --- one's favorite book
aijin 愛人 --- lover
aijou 愛情 --- love, affection
aikenka 愛犬家 --- a dog lover

Words containing "koi"

hatsukoi 初恋 --- first love
hiren 悲恋 --- sad love
koibito 恋人 --- one's boyfriend/girlfriend
koibumi 恋文 --- love letter
koigataki 恋敵 --- a rival in love
koi ni ochiru 恋に落ちる --- to fall in love with
koisuru 恋する --- to be in love with
renai 恋愛 --- love
shitsuren 失恋 --- disappointed love

"Koi" is a love for the opposite sex, or a feeling of longing for a specific person. It can be described as "romantic love" or "passionate love." While "ai" has the same meaning as "koi," it also has a definition of a general feeling of love. "Koi" can be selfish, but "ai" is a real love. Here are some lines that explain them well:　Koi is always wanting. Ai is always giving.