A "Commercial" is an advertisement (although commercial has another different meaning too). It is most commonly used for television adverts, but could be used for other adverts like on YouTube. "promo" is short for a promotion, which can be an advertisement or a special offer. These could be on TV, or any other place where there is advertising. "air date" is usually used in the context of television and refers to when the show or advert "goes live" (starts being shown).

They are somewhat interchangeable but have small nuance differences. For example, "Our company's promo goes live tomorrow." "Our company's commercial goes live tomorrow." Both of these are valid sentences. However, the first implies a "promotional event" - perhaps it's a commercial, but it could also be a live event or combination of "promotional" efforts, including commercial advertisements amongst others. The second implies a TV, YouTube, radio, etc advertisement will be aired. In either case, you'd need more information to determine what exactly the speaker is talking about. :)