Learn Russian: Мы подходим к стойке администратора. - We are approaching the receptionist's counter.

In English, you simply "approach the counter." But in Russian, verbs of motion with the prefix под- (meaning "approach" or "come up to") almost always pair with the preposition к + the Dative case. Think of it as walking up to or towards a specific target.

Why is the word for counter spelled стойке here?

The preposition к strictly requires the Dative case. The dictionary form is стойка (a feminine noun), and feminine nouns ending in -а change their ending to -е in the Dative case.

Why is администратора in the Genitive case?

It shows possession or association, doing the job of the English "'s". You are literally walking up to the "counter of the receptionist". Since администратор is a masculine noun ending in a consonant, it takes an -а ending in the Genitive case.

Does администратор always mean an office "administrator"?

In a corporate context, it can. But in everyday Russian, an администратор is usually a receptionist, a restaurant host, or a hotel front desk clerk. Because of this, стойка администратора is the standard phrase for "reception desk" or "front desk".