Converting to Buddhism is primarily a personal, mental action that is supported by a community.
We become a Buddhist when we acknowledge the Buddha as our fully enlightened teacher, his Dhamma teachings as our guide to happiness, and the Sangha as his fully enlightened disciples who followed that Dhamma to its final goal.
Officially, we do this by reciting the Three Refuge privately or publicly. If done publicly it can be recited by repeating after a monk, a nun, or a trusted lay teacher.
Usually we agree to follow the five lay precepts at the same time.