"My sister is British."

Translation: Moja siostra jest Brytyjką.

[Kulink]

So I've started figuring out some declensions, and i had noticed many sentences that are a direct assignment using the verb 'jest' end in 'em, such as 'jestem kotem' or 'koń jest zwierzęciem'. What is different about 'British' that it does not follow this?

[Jellei]

Your assumption seems to be right (although how widely it works, it's hard for me to say, maybe all such words, maybe there are exceptions), but for masculine and neuter nouns. kotem/lwem/psem/robotnikiem; drzewem/dzieckiem/krzesłem/jajkiem.

If it was your brother who is British, it would work: Mój brat jest Brytyjczykiem. But as the nationality is a noun, and it also takes gender into considerations, its Nominative is Brytyjka, and "Moja siostra jest Brytyjką " (/Angielką/Szkotką/Irlandką/Walijką/Francuzką/Włoszką/Węgierką etc.)

[zagadka314]

I read that "to be" takes Narzędnik, which is usually -em.

Awhile ago I bought this, which has some of the basics about the cases:
https://www.amazon.com/Polish-Verbs-Essentials-Grammar-Oscar/dp/0071597468/ref=sr_1_2?ie=UTF8&qid=1472135950&sr=8-2&keywords=polish+mcgraw+hill

It is very basic but I guess it is nice to read about the cases.

I like to use this site when I am not sure about which case to use:
http://aztekium.pl/przypadki.py?lang=en

[immery]

Narzędnik ends with -em for masculine, and neuter nouns, and -ą for feminine nouns

[zagadka314]

Thats true, I didn't even refer to that. Thank you

[Rimdus]

Why I cannot use "to" instead of "jest" in this case?

[Jellei]

Theoretically you can (added "Moja siostra to Brytyjka" now), but we don't really recommend this construction when you have a person on the left side of "to". It sounds a bit clumsy, more like "My sister = a British woman" than a descriptive statement that "My sister is British". But we do usually accept it. What we don't accept, by the way, are things like "Ona to Brytyjka". If you have a personal pronoun on the left, it just sounds too clumsy.

[alisa488338]

Czy narodowosc w jez.polskim piszy sie z duzej litery?

[Jellei]

Tak, rzeczowniki określające narodowość (Polak, Niemiec, Brytyjka, Ukrainka) piszemy dużą literą.

Przymiotniki (polski, niemiecki, brytyjski, ukraiński) piszemy małą literą.

[Falstaff4]

why is the noun form "Brytyjka" being used here instead of the adjective "brytyjska". Is the literal translation "My sister is a Briton"?

[alik1989]

Yes, indeed. Using the adjective would be a mistake here.

[Arnoldpitt]

British is not a noun it is an adjective. the noun is Briton

[F4yY9kZj]

Is 'Britisher' colloquial (and, perhaps, pejorative)?

[Jellei]

Yes, it seems that it may be rather colloquial and pejorative.

[F4yY9kZj]

And, of course, there's 'Brit' (totally colloquial).