When you look at tutorials for 自己紹介（じこしょうかい; self-introductions) in Japanese, you might see something like:

　はじめまして。私（わたし）の名前（なまえ）は花子（はなこ）です。

　Hajimemashite. Watashi no namae wa Hanako desu.

　Nice to meet you. My name is Hanako.

But here’s the thing: you will rarely hear Japanese speakers introduce themselves this way. (In my mind, this is the classic classroom-style introduction.) Instead, they will say something like:

　はじめまして。山田花子（やまだはなこ）です。

　Hajimemashite. Yamada Hanako desu.

　Nice to meet you. I’m Hanako Yamada.

Two things that are different:

1) The context (that you're introducing yourself) implies that you're talking about your name, so there's no need to say「私（わたし）の名前（なまえ）は」. So, people tend to drop that redundant phrase entirely.

2) Typically, you will use your full name (surname first, then given name).

In formal settings (I also use this when I meet someone for the first time):

　はじめまして。山田花子（やまだはなこ）と申（もう）します。

　Hajimemashite. Yamada Hanako to moushimasu.

　It’s a pleasure to meet you. My name is Hanako Yamada.

The です is replaced with 申（もう）します. To「申（もう）す」is the humble form of「言（い）う 」(to say). Did you know: 申す is where the「もしもし」(“Hello” on the phone) originated from!

If you are looking to practice and polish your 自己紹介（じこしょうかい）, I can be your sounding board! I offer 25-min and 50-min free-form conversation lessons (with a 15-min trial as well).