One of the great natural wonders of the world, the Iguaçu Falls (Portuguese: Cataratas do Iguaçu, Spanish: Cataratas del Iguazú, Tupi: Y Ûasu "big water") are situated near the border of Brazil, Paraguay, and Argentina. The area is on the UNESCO World Heritage List.
Access to the Falls is usually done through one of the three cities in the so-called tri-border between Brazil, Argentina and Paraguay.
The city on the Brazilian side is Foz do Iguaçu - big, but less secure. The town on the Argentine side is called Puerto Iguazu and is small and pretty. Although the falls are between Brazil and Argentina only, Ciudad del Este, the city on the Paraguayan side, is just across the bridge from Brazil. It's a hectic (but exciting) centre for contraband and cheap electronic goods.
Border crossing between these countries is fairly relaxed - authorities assume most people are on a day trip across the border and won't even stamp your passport. So, if you are crossing into another country for good, you will have to persuade them to stamp you passport (otherwise you could run in to problems later). US passport holders may require a visa to visit the Brazilian side of the falls.
On the Brazilian side, Foz do Iguaçu is about seven hours by bus from Curitiba. From Puerto Iguazu there are several daily buses to Buenos Aires and Posadas and from Ciudad del Este there are lots of buses to Asuncion.
There are hourly (or so) buses between Foz do Iguaçu and Puerto Iguazu and Ciudad del Este. There are also hourly (or so) buses between Foz do Iguaçu and the Brazilian side of the falls and twice hourly (or so) between Puerto Iguazu and the Argentine side of the falls. If you want ride busses from one side of the falls to the other, you will have to take 3 separate buses.
It is only a short walk across the Friendship Bridge over the river Parana between Foz do Iguaçu and Ciudad del Este.
Alternatively, you can take a taxi.
The Iguaçu Falls are an awesome sight as tons of water throw themselves over cliffs and the mist rises amongst the jungle. They are taller than Niagara Falls, and twice as wide, for which Eleanor Roosevelt is said to have exclaimed on her first sight of the Falls: "Poor Niagara!"
It is well worth spending a day on each side of the falls.
On the Argentine side (30 AR$ pp, second day for half price if you get your ticket stamped before leaving on the first day, subsequent days are free if you tell them you are staying at the Sheraton) there are a whole series of walkways and trails by the main visitors centre, allowing you to stand right on the edge of the precipice, below some of the waterfalls themselves, see a good overview and take a short boat trip to Isla San Martin below the falls. Wear waterproofs and protect your camera!
There is a free train running up to a 1km-long walkway across the river to stand just back from the main horseshoe of falls - the Devil's Throat (Garganta del Diablo), where the roar and spray are tremendous.
On the Brazilian side (20 R$ pp, cheaper for Brazilian residents) you get an excellent overview of Devil's Throat and the rest of the falls, from both above and below.
Spectacular boat trips can be made under the falls, but are kind of pricy if you take the Safaris offered on the Brazilian side of the Falls. If you go all the way to the "Las Cateratas" station on the Brazilian side, you will be offered e.g. wildwater rafting & abseiling activities at better rates. When you arrive shortly before the activities close for the day or they're just not busy you can easily bargain about the price and get a really good rate!
From the Argentine side:
On the Brazilian side of the falls you can buy souvenirs, but they are very expensive compared to the souvenirs you can buy on the main road of Foz do Iguaçu. There are some very cheap shops along this road, including a gift shop with many hand-crafted crystal goods.
On the Argentina side, there is a really nice hotel that serves a good buffet. You can see the mist coming out of the falls and also generally see toucans and other birds flying around.
On the Brazil side, there's a buffet that stands right next to the throat. The food is good, the view of the river makes for a surreal sight as you know the falls are really close by but you can't really see them other than the mist and the noise.
Excellent food, and probably the nicest restaurant in Puerto Iguazu aesthetically. The Pollo al Diablo (Chicken Devil's Style) is delicious. English is spoken, great for lousy spanish speakers like myself.
At Puerto Iguazu you can go to "La Barranca", nice place near both rivers and the Three Frontiers Landmark (Marco das três fronteira), nice people and live music at weekends. In Foz do Iguaçu there are a few nice nightclubs at the city center.