I do not claim that these songs are the best ever in each of the following languages, but at least I love these songs. All of them. I think each of them can represent their respective language.

Slavic languages
1. Belarussian - Ляпис Трубецкой - Фридом Беларусь
2. Czech - Ivan Mládek - Jožin z bažin
3. Bulgarian - Азис - Мразиш
4. Macedonian - Гоко Танески - Jaс jа имам силата
5. Polish - Maryla Rodowicz - Hej, sokoly
6. Russian - Тайм-Аут - Саша, я хочу стать космонавтом
7. Slovak - Radix - Ešte stále
8. Ukrainian - Олександр Пономарьов - Я люблю тiлкi тебе

Baltic languages
9. Latvian - Brainstorm & Youth choir "Kamer" - Bilde tava
10. Lithuanian - Mango - Taves prašau

Germanic languages
11. German - Eisbrecher - Vergissmeinnicht
12. Danish - Leaether Strip - Smerte
13. Dutch - Blof - Wat zou je doen?
14. Norwegian - Burzum - Hvis lyset tar oss
15. Swedish - Candlemass - Bullfest
16. Yiddish - Ян Табачник - Киевер трамвай

Romance languages
17. Spanish - Los Lobos - Cancion del Mariachi
18. French - Mylene Farmer et Jean-Louis Murat - Regrets
19. Italian - Riccardo Fogli - Storie di tutti giorni
20. Romanian - O-Zone - Printre Nori
21. Latin - Die Form - Cantique

Other Indo-European languages
22. Irish - Enya - The Celts
23. Albanian - Leonora Jakupi - A vritet pafajsia?
24. Persian - Arash - Boro Boro
25. Gypsy - Goran Bregović & No Smoking Orchestra - Bubamara
26. Bengali - Garry Shyman & Palbasha Siddique - Praan

Finno-Ugric languages
27. Escalator - Számítógépés lovagok
28. Estonian - Urban Symphony - Rändajad
29. Finnish - Turmion Kätilöt - Verta ja lihaa
30. Komi - MidWinter Magic - Сьöлöм нöйтчö
31. Mari - Романов Сергей - Салтак
32. Udmurt - Бурановские бабушки - Айшон (Как мне жить?)

Other languages
33. Turkish - Mor ve ötesi - Deli
34. Hebrew - In Extremo - Le 'or chiyuchech
35. Tagalog - Freddie Aguilar - Anak
36. Wolof - Youssou N'Dour & Neneh Cherry - 7 seconds
37. Unknown language - Wise Hand - Manschoud

Artificial languages
38. Esperanto - Dolchamar - Ni cxiuj ni
39. Simlish - Dexter Freebish - Pretty People
40. Unknown language - Wumpscut - Churist Churist
41. Unknown language - Ishtar - O Julissi na jalini

Bonus
Songs in mixed languages