While I don't know about the ATXMega, the 8-bit AVR chips like the ATmega328 used on the newer Arduinos use the AVR-GCC compiler. This allows for compiling C and even C++ to an AVR chip. One level above the AVR-GCC is the AVR Libc, a C library that makes programming for the AVR a higher level task (no longer have to refer to registers directly, and so on).
The Arduino IDE uses AVR-GCC and AVR libc library in the backend. In addition, the Arduino IDE makes other libraries available, like a nice Serial interface.
Finally, the Arduino comes with a bootloader burned on the AVR chip. The bootloader simply makes it possible to program the AVR using a serial connection (from USB) instead of an In-Sytem Programmer or Development Board.
Enough backstory, to answer your question: The Arduino can be programmed in C and even C++. The libraries available are written in C and everything will compiled using AVR-GCC. The Arduino IDE isn't even required.
Edit
There seems to be a decent amount of interest in this topic. I wrote a blog post to try and give more in-depth details on the AVR, Arduino, and AVR-GCC.
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