Ordinal numbers ("first," "second," "third") differ from cardinal numbers ("one," "two," "three") in that they establish order or rank. Forming the ordinal numbers is luckily relatively easy in German. Generally, you will either add -te or -ste on the end, and make sure you have the correct declension.
For numbers below twenty, you simply add -te (zwei — zweite) with the exception of the following ordinal numbers: eins, drei, sieben, and acht.
Die erste Station seines Besuches im September ist Berlin.
The first stop of his visit in September is Berlin.
Caption 2, Der Papst - Hier wohnt der Papst
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Die Nashville-LP „Could Have Been Mine“ [sic, „Could've Been Mine“] ist die dritte Platte der Band.
The Nashville LP "Could Have Been Mine" [sic, "Could've Been Mine"] is the band's third disc.
Caption 43, Ann Doka & Band - New Country aus dem Rhein-Main-Gebiet
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Beim Bauern gibt es noch eine weitere Sonderregel,
With pawns there is yet another special rule,
und zwar: Wenn der Bauer von der siebten auf die achte Linie vorrückt.
and that is: if the pawn advances from the seventh to the eighth line.
Captions 56-57, Schach - mit Jenny - Part 1
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In this last example, you can see the declension based on grammatical case (dative — note the "n"— and then accusative) at work. It follows the same patterns that adjectives do. Note that you may also see "seventh" translated as siebente (in this case it would be siebenten) rather than siebte.
Starting with the cardinal number zwanzig ("twenty"), -ste is added to create the ordinal number.
Man sagt zum Beispiel:
We say, for example:
„der erste Januar“ oder „der vierundzwanzigste Dezember“
"the first of January" or the "twenty-fourth of December"
oder „der dreißigste Februar“, den es nicht gibt.
or "the thirtieth of February," which doesn't exist.
Captions 12-15, Zahlen mit Diane - Ordinalzahlen, Vielfache und Bruchzahlen
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Zu seinem einhundertfünfundneunzigsten Geburtstag
For his one hundred ninety-fifth birthday,
hat der Philosoph seine Geburtsstadt Trier erobert.
the philosopher has conquered his birth city of Trier.
Captions 3-4, „Mini-Marxe“ - In Trier
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It is important to note that, just as "first," "thirteenth," and "twenty-third" would often be abbreviated as "1st," "13th," and "23rd" in English, a period may be used in German to denote ordinal numbers, particularly with dates. The example above would read:
Man sagt zum Beispiel: „der 1. Januar“ oder „der 24. Dezember“ oder „der 30. Februar“, den es nicht gibt.
Further Learning
Take a look at this chart and watch Diane's video in its entirety to get a great overview of this topic. For more on adjective declension, refer to this page.