"My name is John."
this is a humility expression. you use this phrase in the introduction yourself when you are introduced to high rank superior, or when you are a speaker, speech to the audience you first meet at a conference or a lecture.
If you are introduced to friends of your friend, "ジョンです" or "ジョンといいます" is better to use.
と is not only "also/and." It's also the "quoting particle" in Japanese (among other things). You use it to mark the end of a quote that someone says or a thought someone has. In this case, the phrase <X>ともうします loosely can be translated to, "I am called, 'X'. (note the quotes)