やっと is an interesting word. It has elements of both finally and just in its meaning. I think there is actually a better translation for it though. やっと according to jisho.org is translated as
at last; at length. I like these translations better because it gives a sense that your task has taken a lot of time, or that it has taken a significant amount of time longer than you expected.
リポートを書きました。(I wrote my paper)
リポートをやっと書きました。(At last, I wrote my paper.)
Jisho.org also translates やっと as
barely; narrowly; just; by the skin of one's teeth. In the example sentence above we could also apply that meaning as well, and get something to the effect of:
I was barely able to write my paper.
You will find that the translation will vary with context. I'll give an example.
A:リポート書いた？ (Did you write your report?)
B:うん、先週書くつもりだったけど、昨日、やっと終わったよ。 (Yes, I was supposed to write it last week, but I was finally able to finish it yesterday.)
Person B in this case is using the first definition,
at length; at last. We can tell this because it took longer than he/she expected.
Lets continue this interaction between these two people, but two weeks down the road.
Two weeks later...
B: お前もリポートやっと書き始めたんだ？ (#1) (Have you finally started writing your report?)
A: いや、まだ。だって、このクラスあんまり好きじゃないし。(Nope. Actually, I really don't like this class.)
B: でも、明日までには書かないと。宿題をやらないと、先生に… (But, you have until tomorrow to write it. If you don't do your homework the professor will...)
A: 分かってるよ！今晩、書くつもり。(I know, stop reminding me! I'm planning on writing it tonight)
The next day...
A: やばかったけど、おかげでやっと書けた。(#2) (It was tight, but thanks to your help, I barely finished writing the report in the neck of time.)
B: よかった。(Good for you.)
(#1) In the example above is still the first definition.
Have you *finally* started writing your report? I used finally here because it sounds more natural in English than the direct translation, which is
Have you started writing your report at last? We use this translation because it has taken person A a long time to get started writing his report.
(#2) In this example, we get the second definition.
I cut it close, but thanks to you I was just barely able to finish the report in the neck of time.
To put it simply, you use やっと when something has taken a long time (finally), or when you were cutting things close (just barely).