The airport in Ho Chi Minh City is Tan Son Nhat, after the village it was originally built on.
You will find both the international and domestic terminals on the same site.
The airport code for Ho Chi Minh City is SGN, referencing the days when the city was officially named SAIGON.
Tan Son Nhat Airport is only 8 km (~5 miles) from the center of District 1. Even in heavy traffic, it shouldn't take more than 30 to 40 minutes (and 8 about 10 USD) to get to your accommodation if you are staying in District 1.
The government is building a new international airport at Long Thanh, about 50 km (32 miles) from town. But the first stage of this facility come online until at least 2025. (Reportedly) By then the metro will hopefully reach there to make the trip into the center of town a breexe...but we'll see!
The domestic airport will remain at Tan Son Nhat.