Sint Eustatius, locally known as Statia, is a small island in the Caribbean. Politically, it is a "special municipality", fully integrated in the Netherlands proper. The capital, Oranjestad (Dutch for Orange Town) is divided into Lower Town, a single street by the water front, and Upper Town, which is the main part of town. The largest and longest running industries on the island are tourism (mostly dive tourism) and port services for the passing oil tankers from Venezuela.
Statia is small and it's very laid back and quiet. Do not come to this little island expecting nightlife, shopping, etc, or you will be disappointed. Nonetheless the island has considerable charm of its own, assuming you can slow down and appreciate what surrounds you. Although it is a Dutch island, everyone speaks English.
There is a spectacular long sandy beach on the Atlantic coast, but the waves and currents are so strong that it is not safe to swim there. Swimming is possible at the north end of Oranjestad Bay (the more sheltered Caribbean coast) where there is a decent amount of sand.
Get in 
Although Statia is fairly close to Saint Kitts and also not far from Saba, there are no regular connections between those islands. Rarely (offered a few times each year on public holidays) it is possible to take a ferryboat from the island of Nevis to Statia and back again in one day, giving you about 6 hours to spend on Statia.
Most visitors arrive on Statia via the only regularly operated service that flies into Statia's F. D. Roosevelt Airport (IATA:EUX), the regional airline Winair, which services Statia from Sint Maarten. A valid passport is the entry requirement for visitors, including American and Canadian citizens. Visas may be required for passport holders from non- G20 countries.
Be aware that the majority of the island businesses do not accept credit cards. Bring plenty of US dollars. There are only two ATMs on the island, one at the bank and one at the airport. These ATMs run out of money frequently and then are sometimes not refilled for a couple of days.
Statia is on Atlantic time (GMT-4): same as US Eastern Daylight Time, one hour ahead of US Eastern Standard Time.
Get around 
The island is small enough that, assuming you are fit, you could walk or ride a bike just about anywhere you might want to go, but it usually gets hot in the middle of the day, making walking less pleasant. There are a couple of people who run taxi service, but book in advance.
The St. Eustatius Historical Foundation published a very nice little book, available almost everywhere, that outlines a 1-mile walking tour of the capital, Oranjestad. The tour route starts at the port and winds through all the key historical sites. The central district of the capital has a number of very attractive restored buildings, including a museum run by the Historical Foundation. You could pick this book up at the headquarters of STENAPA, which is only a few yards from the port in Lower Town.
Hike The Quill, the island's dormant volcano. Visit the botanical gardens. Go diving or snorkeling; the island is famous for its underwater life.
On land, the animal life includes some large iguanas. An abundance of goats, cows and chickens roam freely. Most of them no longer below to anyone in particular, but are instead remnants of the custom that those who own more animals have more wealth. A concerted effort is being made to round these animals up and perhaps fence them in, because of their environmental impact. Some maybe/maybe not feral cats and dogs also wander about. Do not expect one to curl up in your lap while you read a book, but you might be able to lure an occasional visitor to adopt you by setting some food outside.
In terms of history and archeology, Statia was at one point the most important port in the New World. During the 18th century it rose to prominence through a combination of lax Dutch trading controls and the American Revolution. Successive transfers between Dutch, British and French control compelled the once-prosperous merchant community to seek better profits elsewhere, but the archaeological record records their presence. Hence Statia's current motto, "The Historic Gem." Nowadays, the only visible record of its once-proud presence are the fragments of pottery incorporated into local driveways and a fair amount of crumbling 18th century Caribbean architecture. An archaeological field station run by the Sint Eustatius Center for Archaeological Research (SECAR)  is the only remaining effort to restore the heritage of Statia to American history.
People who are interested in nature should definitely visit the headquarters of STENAPA (the island's national parks foundation) in Lower Town a short walk from the harbor, and talk to the staff there who will be able to suggest the most interesting areas for you to go to.
Wave and say, “Good Morning” and “Good Afternoon” as the locals are very friendly and it is a known custom on the island.
As is the case for almost any beach destination in Florida or the Caribbean, bring insect repellent.
Try the Fruit Tree for local Dutch cuisine. Smoke Alley is where most of the US contractors eat due to their large portions and abundance of American staples (cheese steaks, burgers, etc.) There are five different Chinese food restaurants on the island; Sonny's is the best.
There are only a few places to drink in an island less than 8 square miles in area. The longest running, owner-operated bar is Chuckies. The expatriate community typically restricts itself to this bar as the owner will frequently drink with patrons and keep the bar open until the last customer leaves.
The Old Gin House may be considered the most upscale hotel on the island. Their rates begin at $145 per night. They serve breakfast only, which is included in the room fee, 7:00-10:30. WiFi access is in the lobby and the outdoor patio restaurant/bar.
Kings' Well is a cute resort out by the curve from Upper Town to Lower Town. Very pleasant hosts, and the island macaws visit for the complimentary breakfast. You could eat every meal there if you desired. WiFi on premise. Not much nearby, but short walks get you to the dive centers or the main part of Upper Town.
Go next 
Everyone leaving Statia has to pay a departure fee in US dollars.