After Beco posted about travelling in Europe and good ol' Ceronne about visiting Paris, I just feel compelled to contribute.  However, as many of my readers aren't that proficent in Spanish, English is the language of choice.

Brussels, being between Paris and Amsterdam, can be easily missed when touring Europe.  It is frankly a mistake.  Sometimes berated as a city with no character of its own, the truth is that it has many, as it sits right at the border between Latin & Germanic Europe and is rather appropriately a mix of both.  As such, be aware that even though the city is mostly French-speaking, Dutch is also common and English is widely understood.  Furthermore, don't be surprised if somebody speaks Spanish.

When to arrive?
Preferrably before the weekend.  Pack an umbrella, it tends to rain very often.

How to arrive?

• By plane to the airports at Brussels Airport (Zaventem) or Brussels Airport South (Charleroi).
• By train to its different stations (usually Gare de Midi / Zuidstation or Gare du Nord / Noordstation.)
• By bus (e.g. Eurolines)

From the airport to the city
Quite easy.  Coming from Zaventem there is a train every 20 minutes, which costs 2.8 EUR.  From Charleroi is a little more difficult, there are hourly buses as that airport is quite far away and only serves budget airlines.

Transport within the city
Brussels has two metro lines, buses, trains & trams.  I would recommend buying a 10 trip ticket at the vending machines at the metro stations, since the city is comfortable enough to walk to most places.

• Don't try to take the public transport without a ticket.  Even though It is possible, the fines are quite steep.
• There are only night buses during weekends, so I recommend staying at a hotel within walking distance of downtown.

Accomodation
If you're on a budget, I can recommend Hostel Van Gogh and Sleep Well hostel.  Otherwise, there are plenty of nice little hotels in Avenue / Adolphe Max / Laan or Place Rogier / Rogierplein.

• I recommend checking out otel.com.

Basic tourist information
Preparing yourself

• Map of Brussels.
• Map of the public transportation system (includes bus and metro)

What to see?
Basics

• The Grande Place, one of the most beautiful town squares in Europe. 
• The Manneken Pis, the renowned statue of the peeing kid.
• Just walk around downtown.

Further away

• The Laeken Gardens, including the Japanese Pagoda and the Chinese Pavillion.
• The European Quarter, with the buildings for the Commission, the Parliament and the European institutions.
• The Belgian Comic Museum (Centre Belge de la Bande Desinée).  Required if you're a fan of Tin Tin or Lucky Luke.
• The Parc du Cinquentennaire with the Royal Museums of Art & History and the Military Museum.
• The Atomium and the Mini Europe amusement park.
• The Bois du Cambre Park.  More like a mini-forest, really.

Within 3 hours of the city centre

• Tervuren and its Museum of Central African Cultures
• Antwerp, the centre of Flemish culture.
• Ghent, another nice Flemish City.
• Brugge, the best preserved city in Flanders.  Going there is like being back in the high middle ages.
• The Adrennes forest.

Where to eat, drink & shop in Brussels?
Eat

• One of my favourite places is, without a doubt, De Ultieme Hallucinatie in Rue Royale / Koningsstraat.  Typical Belgian fare in an art déco building.  Not exactly in downtown, though.  Try the Carbonnade á la Flamande.
• If you don't mind eating with other tourists, Rue des Bouchers is a typical haunt offering Belgian & international cuisine, next to the Grande Place.  I recommend the first restaurant to the right, next to an art gallery, but can't remember the name.
• In Rue / Jules van Praet / straat there is plenty of good restaurants.  Le Chat Noir is nice for mussels (only in season), Réves d'Asie has excellent Thai food and there is plenty others.
• Nearby, at Arteveldestraat (I think), there is an excellent Vietnamese haunt called Da Kao II.  Very affordable prices as well.
• If you're in a budget and its lunch time during a weekday, you can have a baguette at a sandwicherie.  There's plenty in the business district around the Gare Centrale / Centraalstation.
• Furthermore, some of the best things in Belgium are edibles: French fries (known here as frites / frieten) and chocolate.  For the fries, the best ones are supposedly at the Maison Antoine at Place Jourdain, whereas the best chocolates are Neuhaus and Pier Marcolini (although Leonidas are also quite OK and much cheaper).

Drink

• Unless you're a teetotaller, a visit to Brussels is not complete without trying the beers.  You can check the Belgian beer review for a wider selection, but if you're in a hurry I'd recommend the following:
  If you like dark ales, nothing I've tasted beats Chimay Brune.  If you're more into clearer brews, I'd say Brigand Blonde or Ciney Blonde are quite good
  (but you'll probably find Leffe Blonde or Duvel more easily).  If you would like to taste fruity beers, Special St. Louis Kriek or Timmermans Kriek are quite nice.  On the other hand, if the taste of Brussels is what you have in mind, Timmermans Gueuze Lambic is what you seek.  And for white wheat beer, I still prefer Hoegaarden.
• Different bars will have different beers, so don't be disheartened if you don't find your favourite one immedately.
• As for the bars themselves, I like Delirium Café, where you find over 2000 beers from all over the world, Celtica for some fun and dancing, Le Corbeau for dancing on tables and Á la Bécasse for a relaxed, typically Belgian atmosphere.  There is also a nice chicha bar in Greepstraat, but I can't remember the name.  It's easy to find as it is the only one with arabic decor and nice sofas in dark red hues.

Shop

• If you like expensive brands, the area around Avenue Louise / Louizalaan is what you're looking for.
• If you like antiques and African handcrafts, you will find them in the Sablon / Zavel neighbourhood.
• Rue Neuve / Nieuwestraat is also lined up with shops, especially clothes.
• Souvenirs are sold in downtown, especially around the Grande Place. There is also a couple of stores that specialise in European-themed goods, one of them near the Manneken Pis.
• There's also the City 2 shopping centre in case you're still missing something.

Important to know

• Tipping is not needed.
• Avoid theareas around the Gare du Nord / Noordstation and Gare du Midi / Zuidstation at night, or preferrably altogether.
• Be careful with pickpockets.  Even thought the problem is not as extended as in Italy, you can still find them.

References

• Wikitravel
• Wikipedia
• Insight Guides

You can also check my pictures about Brussels in Flickr.