Quick answer
Торт is a layer cake, пиро́г a filled pie, пирожо́к a small hand-held one. Choosing wrongly is the classic slip.
Baked
тортtortlayer cake
пиро́гpirógpie, large filled bake
пирожо́кpirozhóksmall filled bun
пече́ньеpechényebiscuits
пря́никpryánikspiced honey biscuit
бу́лочкаbúlochkasweet roll
Sweet things
конфе́тыkonfétychocolates, sweets
шокола́дshokolátchocolate
моро́женоеmorózhenoyeice cream
варе́ньеvarényejam with whole fruit
сгущёнкаsgushchyónkacondensed milk
зефи́рzefírmarshmallow-like sweet
халва́khalváhalva
Talking about it
сла́дкийslátkiysweet
вку́сноfkúsnodelicious
дома́шнийdomáshniyhome-made
на десе́ртna desértfor dessert
With the tea
Хо́чешь торт?Would you like some cake?
Э́то дома́шний пиро́г.This is a home-made pie.
Я не ем сла́дкое.I don’t eat sweet things.
Мне полови́нку, пожа́луйста.Half a piece for me, please.
Моро́женое is an adjective
Literally “the frozen thing” — it declines like an adjective: моро́женое, моро́женого, с моро́женым. The same applies to сла́дкое (sweet things) and второ́е (the main course).
Refusing politely
A Russian host will offer repeatedly. Спаси́бо, я сыт / сыта́ — thank you, I’m full — is the real stop. Я не ем сла́дкое works too, and nobody takes offence. What will not work is a single quiet нет: that is heard as politeness.
Typical mistakes
Я хочу́ пиро́г — meaning a slice of cake
Я хочу́ торт.
торт is the layer cake; пиро́г is a pie.
моро́женый
моро́женое
It is a neuter adjectival noun.
Check yourself
Which word means a layer cake?
Торт — the celebration cake.