Whilst in other countries the main way of phone messaging is by SMS, using the recipient's phone number as the "address", in Japan this is only possible between phones using the same carrier - we use Softbank, so Softbank to Softbank is possible, but not Softbank to AU or Docomo (AU and Docomo being the other major Japanese cell phone carriers).
However, phones in Japan have an email address (with Softbank phones it is firstname.lastname@example.org , where xxxxx is chosen by the phone user like a normal email address ) and it is usual to send messages by email rather than SMS - you can send/receive messages to/from phones of other Japanese carriers, and also to/from computers in Japan and overseas.
Sending email works just like SMS texting except that you put in an email address and not a phone number.
Although you can receive free of charge, in order to send you need to activate the "mail service" - just call 1400 from the phone and press 41# when you hear main menu.
300 yen will be deducted from the Softbank Mobile prepaid balance (and once a month if you are renting long-term).
----Sending email to other phones abroad directly ---
You can in fact send e-mail to US phones from both cell phones and computers, as long as your know which carrier the recipient uses. The major ones are -
---Sending SMS though your computer ----
There are various options for sending SMS texts to phones from computers eg through Skype.