".380" and ".38" in usual conversation in this country mean .380 ACP and .38 Special respectively.
The .380 is a little problem because there are several cartridges of similar size and similar bullet diameter, but
they are not the same cartridge. Don't worry about it too much, buy ammo that matches the markings on the barrel of your gun, and if you have doubts about a salesman's advice, check with someone else before you buy.
Other names for the .380 ACP, none of which are in common usage in the US:
9mm Browning Short, 9mm Kurz, 9mm Corto, and 9X17mm. Anyone that calls the cartridge one of those names in casual conversation is just trying to impress with their uber "knowledge". The time to know the difference is when dealing with foriegn firearms and ammuniton that may be marked that way, or in other technical discussion where that detail may be important.
Here's as good a discussion as any about the choice of .38 Special or .380 ACP for a personal protection firearm.
www.hipowersandhandguns.com/38specialor380acp.htmIn case you haven't figured it out, the "diameter" designation for a cartridge is not necessarily the actual dimension of the bullets used in that ammunition. For a few examples: .223 Remington uses 0.224 inch diameter bullets, .300 Winchester Magnum uses a 0.308 inch diameter bullet, .380 ACP uses a 0.355 inch diameter bullet, .38 Special uses a 0.357 diameter bullet, and so on. [The diameters for cast bullets will most likely be different (one or two thousandths) than these listed for jacketed bullets.] All of this is esoterica to you unless you reload, although it's not necessarily irrelevent as it's important to know the fact so you can keep yourself out of trouble, less important to know the details.
Keep trying every different handgun and caliber you can get your hands on, and don't let anyone try to steer you to a gun that's "just right for you". You will be able to sort that out on your own soon enough. Even if you make a mistake in a handgun purchase it's not the end of the world because you can trade it for another gun without losing your shirt in the deal.