とうとう is an adverb that is used to express that something 'finally' happens. In Japanese, there are a few ways to do this, but
とうとう is one of the more common. Coming from the kanji
到頭（とうとう）, this structure literally means 到 (to arrive at), 頭 (the head of/crux of) something. Due to this, 'finally', and 'after all' are the two most common translations of
とうとう. Like many other adverbs in Japanese,
とうとう is often used at the beginning of a phrase/sentence.
とうとうお前（まえ）が大学生（だいがくせい）か...
So you're finally a college student... (After all this time, have you finally become a college student?)
とうとうロシアに行（い）く。
I will finally go to Russia. (After all this time, I can finally go to Russia)
While
とうとう is written in hiragana the majority of the time, knowing the kanji will help when trying to remember the expression's nuance. That nuance is that something has 'ended', usually after some sort of long journey/effort.
This adverb may be used to express either positive or negative outcomes, as it focuses more on the time required to achieve the result, rather than the result being good or bad.
とうとう夏休（なつやす）みだ！
It's finally summer break! (Good result)
とうとう夏休（なつやす）みが終（お）わる日（ひ）が来（き）た。
The day that summer break ends has finally arrived. (Bad result)
とうとう is not used in situations where you expect something to happen/start, but there has not actually been a period of time in which some sort of effort was made to achieve a result.
A good example of this is the changing of seasons. You might want to say 'finally, it's summer!', but this is not something that anyone has done anything to achieve. Rather, it came naturally. In these cases,
いよいよ will be used, and has the nuance of 'finally', or 'at last' (in relation to something starting/happening).