Learn Russian: Приём двадцатого февраля. - The appointment is on the twentieth of February.

This is the standard Russian way to say on what date something happens. The ordinal number (the day) takes the Genitive case to mean 'on the...'. The month is also in the Genitive case because it translates to 'of February'.

Why is there no preposition like в or на to mean 'on'?

When you use the Genitive case to state a specific date (like двадцатого), the idea of 'on' is already built into the grammar. Adding a preposition here would be incorrect.

Why does двадцатого have the neuter Genitive ending -ого?

When talking about dates, Russian implies the neuter noun число (date or number). So, you are literally saying 'on the twentieth [date] of February'. Because число is neuter, the ordinal number describing it takes the neuter Genitive ending. Also, remember that the г in -ого is pronounced like an English 'v'.

What kind of 'appointment' does приём refer to?

Приём comes from a verb meaning 'to receive' and usually refers to an official appointment, like seeing a doctor, a government official, or a lawyer. If you were just meeting a friend or having a casual get-together, you would use встреча (meeting) instead.