申します（もうします）is the keigo form of 言います meaning "to say" or in this case, "to call". 「と」is a particle that works like a verbal quotation mark. So, saying "__"と申します is the same as saying is "__"と言います. __と申しますis the polite way to introduce yourself. In informal situations, people will often say __と言います or even just __です.

"I am called ____" = ぼくは　＿＿＿＿　と　もうします。　「も＊＊う＊＊します」

I am called ____. ぼくは____ともうします。 【僕は____と申します。】 ぼく○は○____と○もうします。

In addition to their answers, The most common is "私：わたし", and it‘s used in both formal and informal. "僕：ぼく" is mainly used by boys in either formal and informal. "俺：おれ" and "僕：ぼく" are used by men in informal. I'm guessing that you are an adult, and I recommend to say "私は __ と申します” in formal situations. I hope this may help you.