It depends on whether you want to develop applications for a full 
embedded OS (usually Linux), a RTOS (linked together with the 
application into a monolithic binary) or if you'd rather write a "bare 
metal" application which does not use an operating system, but accesses 
the hardware via driver libraries.
Furthermore, there are different ARM families in use, most notably ARM7, 
ARM9, ARM966T and Cortex-M3. These cores are embedded in micros from 
various manufacturers, like Atmel (AT91SAM, AT92RM), NXP (LPC2xxx), STM 
(STR9, STM32) and others.
On the smaller, ARM7-based variants a full OS can not be run due to 
memory restrictions mostly. But they are cheap and easily available if 
you can live with using a RTOS, which allows you task scheduling and 
such, often handles additional things like TCP/IP if available), or 
writing all needed functions by yourself. ARM7-based boards are fairly 
easy to build for the hobbyist, and most solutions are cheap and 
single-chip.
If you want to go all the way, though, you could look out for readily 
available ARM-based hardware, which is not only found in routers. The 
Philips VP6500 IP-Phone is a hackable linux-based ARM platform recently 
discussed here in Germany, for example.