I think I have to be at least 21, right?
I couldn't find any information about this.
Is it possible to accompany a person, who's not 21 yet? Or do I have to be his/her mother/father?
It very much depends where you want to be for the drop.
Supernova hosts a party which is technically in Times Square, just high up, and ages 18+ may attend, or 16+ with an adult.
The Times Square Family Pass lets a family of any ages, although drinks will differ depending on age. Also you'd need to be with your family, presumably.
The Wikihow article also points out that the general access will see elderly and children as well as the usual revellers, but to remember that it's bitterly cold at that time of year. You also need to queue almost the entire day to get in.
Given the thousands of people, the surging crowd, drunkenness, night, the need to queue all day and the need to get back home afterwards, it'd be highly unadvisable for a child/teenager to do it on their own, however, with a family, it's certainly possible, and there are family friendly areas.
For just watching the ball drop and the festivities taking place in Times Square from within Times Square there is no age minimum. You just show up and go through all the police barricades to the "pen" where they put you. You cannot leave and return to the same place, if you leave you have to go to the latest "pen" the police set up.
It's quite chaotic getting in (I see it every year, I live four two blocks west of there) and there may be people around trying to sell fake tickets to get in. You can cut through some of it with tickets or passes to various private parties in restaurants, bars and hotels around there or hotel reservations. The police will ask to see those before they let you through.
And try to get there early if you want to see the ball or the stage early. As the afternoon and evening wears on, the police will be getting people in farther and farther back on Broadway. If you get there after the sun sets you'll be far enough back that you might want to carry binoculars to see.
Like the other answers here say, there's really no age limit for people to be out on the street, and as the night wears on on it gets so hectic and full of people that Times Square, the busiest, most crowded block in NYC becomes literally un-walkable! So there's no age requirement for just "standing on Times Square" but I wouldn't be surprised if lost-looking "kids" are at least questioned by police and asked about their parents since it's madness: thousands of people, drinking champagne on the street, fireworks etc. It definitely looks cool on TV though!
On sort of a random whim last year I decided to take a week off and board an Amtrak train from Atlanta and arrive in New York for New Years. I arrived at Penn Station at 1pm with a bottle of champagne in my hand as I got off the train (I was 25). Looking like a typical tourist, from there all the way to my hostel, every New Yorker I talked to advised me against going to Times Square and told me that it wasn't worth it. I figured it was only because they lived here and had been there lots of time. Maybe so for some, but they were all right. Only two types of locals go there on New Years: crazy New Yorkers, and New Yorkers that work that night at Time Square businesses!
I went and got close but decided to turn back. It bitterly, freezing cold, and you literally are penned in and cannot easily move around. If you don't get there 8-12 hours hours before or you're not in a restaurant or in a private party etc, you're going to be far away, and penned in: You cannot go to the bathroom, you cannot move, you literally just sit there freezing until ~1am when the masses start to try to leave. It takes hours to make it back home too. So I would only recommend it if you can bear the cold and miserableness of queuing and standing for hours and are at least old enough to get into a private bar or something to stay out of the streets. Most other tourists I met that just went to Times Square gun-ho-style like me but stayed said they would never do it again!