Welcome to Gaia! ::

Yeah, there isn't an "alphabet" in Japanese. It's completely phonetic, each sound has its own character. All of the characters shown earlier in the thread are hiragana, which are typically used for grammatical purposes (like, you write "is" in hiragana as です, desu). There's also katakana which is used for foreign words and names, so you'd use that for your name. As said earlier your name would be pronounced as erizabesu, which is written as エリザベス. And then there's also kanji, the cool-looking characters that have meanings, like 火 means fire and 青 means blue. But you should learn hiragana and katakana first before even thinking about kanji.

If you want to learn hiragana, here's a good site to learn it at:
http://smart.fm/lists/24666-hiragana
You'll have to sign up for the site but it takes 10 seconds and you don't get spammed or anything.

You don't see what you possess, a beauty calm and clear
It floods the sky and blurs the darkness like a chandelier
All the light that you possess is skewed by lakes and seas
The shattered surface, so imperfect, is all that you believe

Hiragana (the chart above) is one of the alphabets. The one used for foreign names (and other purposes) is katakana.

Erizabesu = エリザベス (in katakana)

To break it down;

エ e
リ ri
ザ za
ベ be
ス su

I hope I'm not confusing you.
Oh, and the "u" on the end is silent.

A site I used in the beginning was this one. It has the charts, and then you can pick which characters you want to practice and it let's you go through them at your own pace.