The 380 round is known as a 9mm short. No the 9mm luger, parabellum, whatever is not interchangable with the 380.
380 is a round that has ballistics like a 38 special, a common revolver round. 380 ammo does not have enough energy to lock a slide back so they use a blow back design. It is generally a lower recoil than the nine but a lot of that depends on the gun.
380 ammo is expensive compared to the 9mm brother. it is not a bad round at all but many will advise it too weak and too expensive to consider.
now for my unsolicited advice.
I would suggest that rather than getting a 380 to carry, get a j frame smith. load that thing with low recoil hollow point federals for carry and practice with plain old white box winchester or whatever else of quality is on sale.