Finnish and Hungrian are indeed representative of the Finno and Ugric branches of Uralic, the others being the Samoyed languages and Sammic (Lapp). The Turkic languages (Turkish, Chuvash,Turkmen, Uzbik, Azeri, etc etc), along with the Mongolian and Tungus (Manchu, Evenki, etc) lanuages, constitute the Altaic family.
Some scholars believe that there is a Ural-Altaic family that unites these, but the late Joseph Greenberg (IMO the greatest linguist of the 20th century) showed that they are two branches of the EuroAsiatic family, which also includes Indo-European, Japanese, Korean, Ainu, Gilyak, Chukchi-Kamchatkan and Eskimo-Aleut.
BTW, the Celtic languages are one branch of Indo-European; for some reason a lot of people are under the mistaken idea that they're also related to Basque, which is not closely related to anything else.