What's the difference between "cardinal" and "ordinal" as adjectives?

What's the difference between describing a number or place as cardinal versus ordinal? They both seem to have to do with the order of things. For example, when saying "He is the first in his class", is first ordinal or cardinal? What are the usage differences here?

Cardinalis means "that upon which a door hinges, pivotal" in Latin, from cardo, "hinge, pivot". From this it acquired its secondary meaning, "important, principal", which it still has in English (e.g. a cardinal sin). Its third meaning is derived from this: a cardinal number is a "principal" number, i.e. one that simply says how many objects there are.

That'll be one pound.

Two billion people might die.

Ordinalis means "in order of succession, of an order". It comes from ordo, "order, rank". An ordinal number is an adjective that denotes what place an object has in a certain order. The names of the ordinal numbers are usually derived from the cardinal numbers by adding -th.

That is my second victory.

This is the tenth time she's dumped me.

While a cardinal number refers to several objects ("three apples"), an ordinal number refers to only one of those ("the third apple"). An ordinal number is hence dependent on the notion of a cardinal number: there can't be a third apple unless there are at least three apples. By contrast, there can be three apples without one being the third, if they are just not arranged in any particular order.

The Romans used these terms the same way. They also had distributive numbers, which indicated "every third apple", or apples "in triads", "three each".

Cardinal has to do with cardinality or the magnitude or quantity of things.

Ordinal has to do with ordinality or the ordering or ranking of things.

Thus, first is an ordinal number. Its cardinal equivalent is one.