Heedless,

The way you've described this, it's not possible for two reasons:

1) arduino, as far as I know, is not capable of functioning as a USB host.

2) processing doesn't run on the arduino, it runs on a PC and can communicate with the arduino via serial (or any other compatible protocol).

I know that some people have created ethernet capable ardiuno boards, but I don't think they were wireless.  I also think they were just using UDP, not TCP/IP.  You'd have to do some googling, or try the arduino forums for more detailed hardware stuff.  Perhaps someone out there has created a wifi module that can interface with arduino easily...

-Ben

Yes, that's what it means.  The arduino can communicate with a PC but isn't capable of running applications like processing, flash, or Max.

To write programs that run on the arduino, you need to use the arduino development tool (which looks similar to processing):

http://www.arduino.cc/en/Main/Software

With that you write C code that gets compiled into a format that the little microprocessor can understand.  So, for example when you want to have processing and arduino work together, you actually have to create TWO separate programs that talk to each other.  One runs on the PC, and one on the arduino.

hi,

arduino is run by an atmel microcontroller. this means 8 / 16 Mhz of clock speed, and some kB of memory. Should be obvious that this doesn't run java You program the Arduino in C, through its IDE. The Arduino IDE looks similar to the Processing IDE, also the 'simplified C' used in Arduino is quite similar to the simplified Java used in processing. The Arduino can run indipendent from a computer.

It is however possible to connect Microcontrollers to the web, but you'll need some additional hardware, like some serial2ethernet modules. A mature part of Tom Igoe's "Making Things Talk" is just about this topic.
You'll most probably not be able to use the wifi sticks you already have.

For the application you mentioned, you'll probably need a setup like this:

Arduino (has a Led to show 'got mail')
 - communicates to
Lantronix Embedded Device Server (or XPort, or WiMicro)
 - calls the a php script
PHP script on a webserver (checks for new mail)
 - answers back to the Lantronix if we have new mail (true/false)

Yes, it's possible to have your arduino access the internet:

http://www.ladyada.net/make/eshield/

if you want to do it wirelessly, that's more tricky, try doing it over xbee (note that you will probably use the xbee to connect wirelessly to another computer that has internet - as far as i know, there is no xbee -> internet direct interface, but i don't know much about xbee anyways =[):
http://www.arduino.cc/en/Main/ArduinoXbeeShield

if you want small and real wifi, you can try:
http://www.gumstix.com/
but that's a whole 'nother story


for your purposes, i'd try the ladyada's ethernet shield - looks very promising and i'll probably try it myself sometime - i seem to remember that you can access your gmail really easily by using the RSS feed at:

http://mail.google.com/mail/feed/atom

I think you can authenticate in the form:

http://USER:PASSWORD@mail.google.com/mail/feed/atom

with that ":" between the USER and PASSWORD - someone correct me if i'm wrong

(oh, and you might want to use https:// because i think the one i just posted sends your request in cleartext (disclaimer - the https:// might send your request in cleartext anyways, but iono))