The (1)555-555-1234 system is unknown in Europe. They don't have the same number of digits for all phone numbers like in the US.
Example: A number in Berlin (area code 030) would be 030-123 456 789, a number in Hintertupfing, Bavaria could be 087542-231.
If somebody in Berlin would call above number within Berlin, he'd dial 123 456 789; people outside of Berlin, but still in Germany, would dial 030 123 456 789, and people in the US would have to dial 011 49 30 123 456 789.
Cellphones are an entirely different matter: Somebody's cellphone would have the number 0171-12345678, and 0171 would be the "area" code for all the cellphones for one provider. Or to put it the other way around: if you see a German phone number that has an area code beginning with 017, it's definetely a cellphone, and you'll have to pay the extra charges (incoming calls are free for the cellphone user), and you won't know in which area the callee resides.
It's quite similar in all European countries except France.
So, in order to place your call to Southampton, you'd have to dial:
01144 2380 666 805
(You grouped your number wrong, that might have confused matters)
011 Dial into the international system
44 into Britain
2380 Southampton
666 805 your party