Niko niko (にこにこ) describes a warm, happy smile that stays on someone's face.
Some people seem to be smiling whenever you look at them. Not laughing, not grinning: just a soft, happy face. Japanese has a word for exactly that.
What it means
にこにこ is a warm smile that stays. It is silent, it is friendly, and it tells you the person is in a good mood. A shopkeeper who is glad to see you, a baby who has just woken up, a friend with good news.
How to use it
You usually see it on someone else's face, so the natural form is しています: にこにこしています, “is smiling”. You can also put it in front of わらいます to say how someone smiles: にこにこ わらいます.

ゆきさん、きょうは にこにこして いますね。
Yuki, you're all smiles today.

はい。きのう、ともだちに プレゼントを もらいました。
Yes. I got a present from a friend yesterday.

いいですね!
How nice!
Close cousins
Japanese has a different word for each kind of smile and laugh. にやにや is a smirk: someone is pleased with a secret, and it is not a compliment. くすくす is a giggle you try to hide. げらげら is loud, open-mouthed laughing. にっこり is one single bright smile. にこにこ is the gentle one that does not stop.
In manga and anime
ニコニコ floats beside a character who is beaming. Watch the rest of the panel, though: an artist sometimes draws it next to a character who is smiling a little too calmly, and then everyone else in the scene starts to look worried.
Can you use it?
Yes, and it is a kind thing to say about someone: いつも にこにこして いますね, “you are always smiling”. Just be careful not to mix it up with にやにや, which means something much less friendly.