When saying what my name is, which one should I use and why?
There are some exceptions, but, as a rule, in absence of prepositions, when you translate "X foobars Y with Z", X is in nominative, Y is in accusative, Z is in instrumental.
In "they call me", "me" is Y, so you need accusative of "я", which is "меня".
"мои" is plural of "my" or "mine" and I don't understand why you think it applies here.
Меня зовут is directly translated like
my name is. I.e.
меня зовут Петр ->
my name is Peter. In most cases you should use this construction.
Мои зовут is not a common and grammatically correct construction in Russian language. It can be used as a slang phrase like
my [friends] are calling me. Cant find a proper example in English.
P.S. Of course there are so many cases where you can hear these phrases in a different meanings, but these two are the most common and correct IMO.