Äntligen means 'finally' or 'at last', and is used to express happiness or relief that something has happened or come to an end. For example, you might be pleased to have reached the end of a long week at work, or relieved that Sweden is getting a government after 131 days of waiting.
It's an emphatic word, and can show positive emotion at an event finally being here, or frustration at how long it's taken to get there, or even a mix of the two.
One context in which you might hear it is during the Nobel Literature Prize announcement. Swedish journalist Gert Fylking started an unofficial tradition of shouting out “äntligen!” when the winner's name was called, as a way of poking fun at the fact the winners were often far from being household names. He was eventually banned from the ceremonies because of this, but look out for the in-joke on social media or if you're watching the announcement with Swedes.
Besides heckling prize announcements, there are other times when you should avoid using äntligen.
It might be helpful for English-speakers to think of äntligen as meaning 'at last', because 'finally' is often translated in a different way.
If you want to say 'finally' in the sense of 'lastly', to introduce your final point in an argument or the final stage of a process, you should usually say slutligen or till slut. And if you're saying 'finally/at last' in a neutral way, emphasizing the fact that a process came to an end but not expressing any emotion, the best phrase is till slut, which you can also think of as meaning 'in the end'.
To see how this can make a difference, take the phrase Sveriges riksdag har valt en ny statsminister (Sweden's parliament has chosen a new prime minister).
If you insert till slut – Sveriges riksdag har till slut valt en ny statsminister (Sweden's parliament has, in the end, chosen a new prime minister) – it's a neutral statement emphasizing the time the process has taken and the fact that it's now at an end. But if you insert äntligen – Sveriges riksdagen har äntligen valt en ny statsminister (Sweden's parliament has finally chosen a new prime minister) – you're emphasizing your personal response to the fact that the process is over. Whether that's relief, joy, or something else would depend on the tone.
Another example: if you say till slut somnade Mathias (in the end, he fell asleep), you're saying that falling asleep was the last thing Mathias did, whereas in the sentence äntligen somnade Mathias, the focus is on an emotive response to the fact that Mathias finally fell asleep, perhaps after a difficult day or after having trouble dozing off.
Friday at last!
Sverige har äntligen fått en ny regering efter att ha väntat i 131 dagar
Sweden has finally got a new government after waiting for 131 days