Learn Russian: Наш отъезд двадцать первого июня. - Our departure is on the twenty-first of June.

In Russian, to say that something happens on a specific date, you put the date into the Genitive case. The word for "date" (число) is neuter and is usually left out, but the ordinal number still takes the neuter Genitive ending: первого.

Why doesn't the word двадцать (twenty) change its ending?

When you have a compound ordinal number (a number made of multiple words, like 21st, 134th, etc.), only the very last word changes its form. Двадцать stays in its basic dictionary form, while only первого changes to the Genitive case.

Where is the word for "on" in this sentence?

You don't need a preposition for "on" when giving a full date. Simply putting the ordinal number and the month into the Genitive case (двадцать первого июня) automatically translates to "on the 21st of June."

The word отъезд (departure) looks a lot like a verb of motion. Are they related?

Yes! It is a noun built from the verb family for "going by vehicle" (ехать). The prefix от- indicates moving away or departing. You'll see the exact same pieces in the verb отъехать (to drive away / to depart).