Here's the top three:

1. Visit the old quarter around Lipscani street (a network of pedestrian streets with mid/late 19th century buildings, several even older churches, small stores, dozens of restaurants, cafes, bars etc. )

2. Visit the Palace of Parliament ( one of the world's largest buildings )

3. Go see the Village Museum ( an outdoor exhibition of traditional houses gathered from villages all around the country, located in Herastrau park)

Depends on your interests, as all travel questions do. Also important are the number of hours or days you'll be in the city. I'm assuming you're either on a Danube cruise or perhaps a Black Sea cruise... and will either be coming from Constant(s)a on the Black Sea or one of the Danube ports (which will be a much shorter bus ride to Buchares(h)t.

Either way, or even if you'll be touring on your own...I'll just list my choices, which can be visited within a one full day time frame:

The cute little Palace of Parliament that Nicky Ceaus(h)escu built, a gaudy shrine to a mad man. You can take a tour to see the monstrosity.

The nicely designed parks...close-in lush Cis(h)migiu, with a small swan filled lake, or a little further out, Herastrau Park and it's large lakes. A short way from there, across the lake, is the Village Museum with a broad array of typical rural buildings you would come across if you were to travel the countryside. (I Ihope you have a chance someday)
Another museum you may want to visit is the excellent National Art Museum, with Romanian art largely unfamiliar to art lovers in other parts of the world.

You'll find the city in a certain state of repair, broken pavements, decaying buildings, but for the most part it retains much of it's pre-WWII splendor in the Parisienne tradition..even a copy of the Arc de Triunf near the Village Museum.

A stroll down main-street Calea Victoriei will give you an idea of old Buchares(h)t("Paris of the East")...with its shops and turn-of-the-(20th)century buildings. While the oil fields of Ploes(h)ti about 80kms north of town, were smashed to smithereens by US bombing groups from North African bases, the captal remained quite untouched by the war...although Romanian fascists were responsible for constant bloodshed during those days. In particular, the large Jewish population historically suffered, as prime fascist targets.

Maybe the purists won't agree, but I would suggest lunch at the venerable Carul cu Berre (The Beer Cart)on Str, Starvropoleos...to get a look at old Romania. With more time I would suggest other popular eateries.

If it's one day, you'll keep busy trekking and taxiing all over town. The bus system is a good one as is the metro..stations all over the city.

If you need a hotel, I can recommend from personal experience, The Hotel Opera, well-situated, quiet street, elevator, dining, etc.

Whatever your plans, enjoy a very different European city.

stu tower (I have photos ranging from the 1970's to 2005, if interested)

Thanks for the reminder Flat4...Lipiscani was once the large Jewish district of town..and near the area are several associated landmarks if you are at all interested...the handsome Choral Temple on Strada Street being one...once the headquarters of the late Grand Rabbi of Romania, Moshe Rosen.

The Jewish district was a bit south-east of the Lipscani area but unfortunately it was mostly demolished in the 1980s.
Only the small area approximately around part of Sfanta Vineri street (with the Choral Temple) survives intact but there are a few other sites (the Great Synagogue, the Jewish Museum) hidden behind tower blocks further south.

LBTyler, tower and flat4 gave you a good list of places to visit. I would add a visit to Ateneu (Athenaeum) - the splendid concert hall. If you do not go to a concert, you can visit it after 1 p.m. It will take 30 minutes and you will see one of the most beautiful concert halls in Europe. See http://tour.fge.org.ro/

I agree with Tower, it will be a nice experience to have a lunch or dinner at Carul cu bere. Not only for menu of traditional Romanian food, but more for the interior decoration.
http://www.youtube.com/watch?v=m3rKFL8B9Aw&feature=related

Tower, "Strada" means in Romanian "Street". The address of Choral Temple is Sfanta Vineri Street (in translation, this means St. Friday Street).

I am sorry, Tower, but I think you made a mistake. Never Lipscani was a Jewish district. It was (and it is ) full of small shops. It was named after Leipzig (translated as Lipsca in old Romanian language Romanian). Lipscani mean traders who bring their products (to be sold) from Western Europe.

Speaking about Jewish districts, main were in Calea Vacaresti, Dudesti, Vitan and Dristor.

Vacareshti is the site of the old Tailor's synagogue where I in the 1980's interviewed Iancu Livadaru the "curator", Valtor, but my travels through Romania began in the 1970's. In Bucharesht my several interviews with Rabbi Rosen always indicated that the Lipscani area was part of the old Jewish district. And yes, I know very well that Strada (Str.)is the Romanian for street...Among others, Livadaru and Rosen appear in one of my books based on some ugly Romanian history. But none of that is germane to the OP's question. I could have used your help back then, but actually I had all the help I ever needed from the local Jewish communities from Buch to Cluj to Yash (Iasi), to Sighet(ul), to Birlad and Galatz.

Sorry if I and some other some poster thought it was necessary to hijack your post, LB. I hope you have enough to make your stay, however brief, a happy one.