Tokyo Skytree was opened back in 2012, and takes the impressive honor of being the world's tallest tower (standing at 634 meters) -- and not just that, but it's the second tallest structure in the entire world (only beaten by the Burj Khalifa in Dubai).
The naming of the tower actually occurred through a nationwide vote - there were other options suggested, such as Tokyo Edo Tower, Or Mirai Tower (Mirai means future) but the popular vote went to Tokyo Skytree.
Tokyo Skytree actually has two separate observation decks - one is at 350m high (called the Tokyo Skytree Tembo Deck), and the other is at 450m (the Tokyo Skytree Tembo Galleria). You can either buy a ticket to just go to the first observation deck, or one that covers both. I visited Skytree with my sister when she vacationed here, and we thought that the 350m deck gave us a great view, so we just stuck with that. Same day tickets to the Tembo Deck will set you back ¥2,060, and it's an extra ¥1,030 if you also want to head up to the Tembo Galleria - so if you just do the lower deck you'll save yourself some money, too. But by all means - if you want to get the higher view as well, go for it!
Also, can we just take a moment to appreciate what is probably one of the scariest jobs in the world -- at least to me, anyway! Being a window cleaner at Tokyo Skytree certainly would be an interesting career field, but not one for those who are afraid of heights!
There are comprehensive instructions on the Tokyo Skytree website on how to get there - you can check it out at the link here. If you're heading out to Skytree, there's some great shopping in the area, and Asakusa is not far away at all. You could make a day out of exploring that part of the city, and check out the nearby Sensoji Temple (Tokyo's oldest temple) and Nakamise Dori which is a shopping street that leads up to the temple.