#iimasu

if there's no honorific (san, kun, etc) and it's a formal sentence then they're probbaly talking about themselves.

いいます is the polite form of 言う 「いう」 which means to say. Therefore マリアといいます would roughly be "They say/call her as Maria."

名前 「なまえ」means name. So using this would be a more direct translation of "One's name is..." 「私の名前はジェイソンです。」 の is possession marker similar to "'s" in English. :)

と is often considered as a quoting particle.

For example, 「日本語は難しいと思う」(nihongo wa muzukashii to omou) means "I think that Japanese is difficult." 思う means "to think", so と identifies the bit before it as "the stuff that I think." I hope that makes sense.