Gara de Nord is Bucharest's only major railway station. All international - and most local - trains arrive and depart from here. It is located close to the city centre, one stop west from Piata Victoriei on the M1 metro line.
It is not being too unkind to suggest that the station has seen better days, and that it is not the friendliest place in the city. There is a perennial cast of drunks, tramps, chancers and outright fraudsters to deal with, a situation not helped by an almost complete lack of official information points. As an introduction to Bucharest it's really rather grim, although somewhat in keeping with other railway stations around the country.
Fortunately, Gara de Nord is relatively small. You will not need to spend any real amount of time here. All of the platforms lead to the same main concourse, where you will see signs for buses, taxis and the metro. There are a number of press/refreshment kiosks, a supermarket and even a McDonalds and a KFC. There are public toilets next to McDonalds (which are reasonably if not impeccably clean but certainly usable; the gents at least: they cost 1.50 lei). Avoid all of the currency exchange kiosks - they offer appalling rates of exchange and/or take huge commissions. Use one of the many ATMs to get your hands on local cash instead. There are left luggage lockers (which include plugs where you can charge phones and such like) in the First Class ticket office. They cost from 13 lei (for eight hours) to 20 lei (for 24 hours).
To get to the city centre, take an honest taxi from the taxi station outside (ignore anyone who approaches you inside the station claiming to be a taxi driver). Always double check that you get into a taxi charging no more than 1.39 lei per kilometre: a trip to anywhere in the city centre should never cost more than 10 lei. You can also hop on the metro: you are just one stop away from Piata Victoriei, or take the bus: No. 123 goes to Piata Unirii, No. 85 to Piata Universitatii and No. 780 goes to the airport. Tickets (in the form of an Activ or Multiplu card) can be bought from the little metal kiosk next to the bus stop (which is itself next to the taxi rank). Note that both the last buses and the last metro services leave Gara de Nord at around 23:00.
When departing from Gara de Nord, note that queues at the ticket office can be long: do not leave buying a ticket until the last minute, as you may end up missing your train. Indeed, you are advised to buy train tickets well in advance, especially if you are planning on taking a sleeper train.
None of Bucharest’s reputable hostels send people to Gara de Nord to speculatively meet trains. Anyone who approaches you (and if you have just got off the Budapest train and have a backpack, you will be approached) is trying to scam you: they will try and divert you from your intended destination (often by saying that the place you want to go to is closed) towards another hostel or hotel that they will recommend to you. Just ignore anything they say and wave them away.