The STIB (in French) or MIVB (in Dutch) is the public transportation company in Brussels. They offer three different forms of transportation: the metro, the tram and the bus.
The metro network is really easy to figure out. It's essentially two lines (one going around the boulevards that are the limits of the city of Brussels itself (the Pentagone) and one crossing the city from east to west with two branches at each extremity.
The Line 1A starts at Belgium's biggest stadium: King Baudouin and stops at : Heysel (for the Atomium, Brupark and Heysel exhibition grounds). It joins line 1B at Beekant and among useful stops: Ste-Catherine, de Brouckere (for rue Neuve, Place de Brouckere and Grand'Place), Gare Centrale (Central station), Parc (for the Royal Park), Arts/Lois for the ministeries and embassies, Schuman for the European District. It splits again from 1B at Merode.
The Line 1B starts at Erasme hospital, one of Brussels's biggest universty hospital, it then joins 1 a to cross the city centre and splits at Merode, Alma is the stop you want for the medicine campus of the University of Louvain-La-Neuve and the St-Luc University hospital.
The end of the line is at Stockel.
Line 2 almost completely circle the city.Important stops are Rogier (for rue Neuve and hotels around Place Rogier), you can catch the tram there), Botanique (for the Botanique Garden), Arts/Loi (where Line 1a/b crosses Line 2), Porte de Namur (to access Matonge, the African neighborhood and chaussee d'Ixelles shopping), Louise (for luxury shopping on avenue Louise and hotels), Porte de Hal (for those nightflies who wants to go to the Fuse or eat at Le Bazaar or the early birds who feel like walking to Place du Jeu-de-balles for the flea market) and Gare du Midi (Midi Station), where you an catch Thalys and Eurostar trains.
Crossing the middle of the city is the underground portion of the tram line (also called "pré-métro"), very useful and busy. The important stops: Gare du Nord (North Station), Rogier (for rue Neuve), de Brouckere (where you can catch the metro), Bourse (for the stock exchange and Grand'Place) and Gare du Midi (Midi station).
But the above ground tram is really a great way to explore the city as it has its own space on the road and little affected by traffic. Amongst the interesting lines: Line 92 starting North at the Schaerbeek Train station, crossing the Upper-Town (stops at the Museums of Fine Arts, Royal Square, Petit Sablon, Palace of Justice, Avenue Louise (you can catch the Line 2 metro), Place Stephanie and crossing St-Gilles before reaching the affluent borough of Uccle and the end of the line at Fort-Jaco, close to the Foret de Soignes (Soignes Forest) where many Bruxellois spend their Sundays.
Line 93 has the same itinerary as Line 92 except it splits at Place Stephanie to go towards the Universite Libre de Bruxelles (ULB), Brussels's biggest university.
Line 81 is a nice alternative to go to the Heysel if you don't want to spend your time underground. It is longer but will take you to different neighbourhoods. You can catch it in the underground tram lines stops.
The Line 44 is also important: it is located in the other underground portion of the tram system and you can catch it inside the Montgomery station (Line 1B) and it will take you through some fancy neighbourhoods all the way to suburban Tervueren and the Museum of Central Africa.
The third transportation option is the bus.
There are a lot of lines so I'll only point out 1: the Line 71. A very famous "double-bus" line that crosses the most important area in the center of Brussels before reaching Ixelles and the University campuses.
Line 71 starts at Place de Brouckere and stops by the Galleries St-Hubert, Central station, Place Royale (for the museums of Fine Arts and the Music Instruments museums), Parc Royal and Royal Palace, and Porte de Namur (you can reach Avenue Louise with a short walk) before taking the Chaussee d'Ixelles (some good shopping to do there or you can visit Matonge, the Congolese neighborhood) as it gets deeper into Ixelles, you'll pass by l'Amour Fou (a very popular bar), La Maison de la Radio (Radio House) a recently renovated building that looks like an ocean liner and used to house the first years of the belgian National radio. It is now a cultural center where you can see concerts and the bar has become quite popular. Just after is the stop of les Etangs d'Ixelles (Ixelles Ponds), two ponds that are a little oasis (many students come here to relax, you'll also see families on Sundays and elderly ladies feeding the ducks). The tram carries on to the main University campus (or Campus Solbosch) then, to the Campus shared by the ULB and VUB (its Flemish counterpart) or Campus de La Plaine and finishing its run at Delta where you can catch the metro.
Line 71 is now famous because it is the first line in Brussels that has a night service: the N71. It only works on Friday nights though and until 3:00 am, but for a city that never had night services, it's a start.
A little useful tip... Brussels has no night public transportation services. The last Metro leaves around 1 AM and Line N71 is only one line that works only on Friday. Your only alternative is a taxi (you HAVE to have the number of a taxi company in handy to pick you up).
So, what to do if you're too broke for a costly cab? Well... here's a tip that might be helpful. At night, buses run between their garages and if you know where the bus stops and if you have a ticket, you can take it. To get a schedule, you can go to one of those garages (one of them being in Ixelles, tale the bus 71 and ask for "le depot d'Ixelles". Make sure with the driver that it stops close to where you want to go (the itinerary might be different). One night, I was at the Campus de la Plaine after a party and no one to drive me home. Some friends told me one of those buses was going to drive by soon and indeed, I lucked out! After 10-15 minutes, here comes the bus. It was going to take me close to home but not before doing a loooong circle around Brussels' boroughs. The campus is south of the city and the bus went all the way to Laeken (up north) before driving by the northern edge of inner Brussels. It took an hour, by I was home. Funny story... after 15 minutes, I really, really had to pee. Yes, too much beers will do that to you. I tried to hold it and spend half-an-hour in a torture state I can't even imagine! The little occasional bumps on the road were not making things easier... and even I stopped, where would I go. I almost had lost hope when I see a park. Shame vanished in front of necessity and I asked the driver: "Sir, could you stop 2 minutes. I really have to go... real bad and there's a park there". I must have looked miserable because instead of laughing, the driver gave me a concerned look and asked me kindly. "Do you want me to wait for you?" "That would be kind of you, yes". I ran as fast as I could, desperately looked for a bush tall enough to hide me and relieved myself. It never felt so good! I hoped on the bus again and started striking a conversation with the bus driver. Soon enough, I was left 100 meters from home.
Only at night...
Website for public transportation: