From Namiko Abe,
Your Guide to Japanese Language.
Dialogue in Romaji
Namiko: Hajimemashite, Namiko desu.
Paul: Hajimemashite, Paul desu.
Dialogue in Japanese
Wa is a particle which is like English prepositions, but always comes after nouns. Desu（です） is a topic marker and can be translated as "is" or "are". It also acts as an equal sign.
Watashi wa Namiko desu.
私は奈美子です。 I am Namiko.
Kore wa hon desu.
これは本です。 This is a book.
Japanese often omit the topic when it is obvious to the other person.
When introducing yourself, "watashi wa（私は）" can be omitted. It will sound more natural to a Japanese person. In a conversation, "watashi（私）" is rarely used. "Anata（あなた)" which means you is similarly avoided.
"Hajimemashite（はじめまして）" is used when meeting a person for the first time. "Hajimeru（はじめる）" is the verb which means "to begin". "Douzo yoroshiku（どうぞよろしく）" is used when you introduce yourself, and other times when you are asking a favor of someone.
Besides family or close friends, Japanese are rarely addressed by their given names. If you go to Japan as a student, people will probably address you by your first name, but if you go there on business, it is better to introduce yourself with your last name. (in this situation, Japanese never introduce themselves with their first name).
Translation for the Dialogue
Namiko: How do you do? I'm Namiko.
Nice to meet you.
Paul: How do you do? I'm Paul.
Nice to meet you.
* Katakana is used for foreign names, places and words. If you are not Japanese, your name can be written in katakana.
* When introducing yourself, the bow (ojigi) is preferred to a handshake. Ojigi is an essential part of daily Japanese life. If you live in Japan for a long time, you will begin bowing automatically. You might even bow when you are talking on the phone (like many Japanese do)!