How do you say "congratulations" in Russian?

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Поздравляю!
 [paz-drav-lya-yoo]

Taken from:
http://learningrussian.net/congratulations_verbs.php

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you can say "мои поздравления" [mai paz-drav-'le-ni-ya]

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nb. поздравляю is a verb. я поздравляю тебя - I congratulate you. 
strictly speaking, congratulations is поздравления - paz-drav-'le-ni-ya (noun in plural)

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"Прими мои поздравления! "  (  PRIMI  MOI  PO-ZDRAV-LYE-NIYA )
" Искренне рад за тебя! Поздравляю ! "   (  IS-KRYEN-NYE   RAD  ZA  TYE-BYA !  PO-ZDRAV-LYA-YU ! )

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Usually more correct is "Поздравляю" / "Поздравляем" (as verb, for example: "Поздравляем с приобретением") or "Поздравление" / "Поздравления" (as noun).