"My name is Vera."
Translation:Меня зовут Вера.
Literally, меня зовут = "(they) call me", which is translated idiomatically in the English "I am called".
In this sentence, "me" is the direct object of the verb "call", and in Russian, direct objects of verbs have to be placed ("cast") in Accusative case - and меня is the accusative case form of я = "I".
(In English, we say the "me" form of "I" is in the Objective case, since it's used for direct and indirect objects of verbs, objects of prepositions, and a few other usages.)
зовут is 3rd person plural present tense of the infinitive звать = "to call". The present conjugation of звать is:
Он/Она/Они зовёт Мы зовём
(The accents over the letters come from the conjugation table I copied, and apparently show which syllable is stressed.)
There are parallel instances of similar usages in other languages. In Romance languages (French, Spnaish, Italian, Portuguese, maybe even Romanian), the phrase is usually "I call myself" instead of "they call me", but the idea is the same.
'Меня' is both genitive AND accustive. In this case, we are using the accustive for as the sentence literally translates to "(They) Call me Vera". Where 'they' (which is omitted) is the subject, 'call/зовут' is the verb referring back to the subject and 'me/меня'is the direct object taking on the accustive case!