There are two ways of saying "finally" in French: finalement and enfin. Though they have the same translation and are often used interchangeably in casual speech, these two words aren't exactly synonymous. There's a subtle difference between them that's illustrated in these two examples:
Le grand jour est enfin arrivé.
The big day finally arrived.
Cap. 28, Le Jour où tout a basculé: Nos bébés ont été échangés... - Part 7
Au début... j'étais braquée. J'avais pas envie. Puis finalement j'ai compris que c'était pour mon bien.
In the beginning... I was dead against it. I didn't want to. Then finally I understood that it was for my own good.
Cap. 6-7, Le Jour où tout a basculé: Nos bébés ont été échangés... - Part 8
When you say that something has finally arrived, you're implying that you've been expecting it to arrive for a while. But if you finally understand that something is for your own good after being dead against it, you're implying that you didn't expect to have this reversal of opinion. This is the fundamental difference between enfin and finalement: while enfin describes a foreseeable outcome, finalement describes an unforeseeable one.
Let's look at another example. If you say to someone, je suis enchanté(e) de vous rencontrer enfin (I'm glad to finally meet you), you're saying that you've been wanting to meet them for a long time. But if you say, je suis enchanté(e) de vous rencontrer finalement, you're giving the impression that you didn't really want to meet the person at first, but now you're happy that you did. Which is to say that you shouldn't use finalement in this case, unless you want to hurt their feelings!
Finalement can also mean "in the end," which also has the sense of something not turning out as expected:
Alors demain, finalement, on ira pas au château.
So tomorrow, in the end, we won't go to the castle.
Cap. 54, Le Mans TV: Mon Village - Malicorne - Part 3
Another way of translating that caption would be, "So tomorrow we won't go to the castle after all."
Enfin is used very often in informal speech as a sort of filler word that can mean anything from "well" to "I mean" to "in any case":
Il y en a eu tant que ça? -Oui, enfin, non, euh... quelques-uns, quoi.
Have there been that many? -Yes, well [or "I mean"], no, uh... a few, you know.
Cap. 37-38, Le Jour où tout a basculé: Notre appartement est hanté - Part 2
...où nous sommes au métro Jaurès, enfin, où Paris-Plage a accès à l'eau.
...where we are at the Jaurès subway stop, in any case, where "Paris-Plage" has access to the water.
Cap. 2-3, Lionel L: Paris-Plage - Part 2
Enfin can also come in handy when expressing impatience or frustration:
Mais enfin, relève-toi!
Come on, stand up!
Cap. 2, Il était une fois - les Explorateurs: 15. Bruce et les sources du Nil - Part 3
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