My knowledge of what the teensy is is very limited, and I would love a brief explanation of it.
mem has done a great job, but I'd like to emphasis the thing that makes Teensy interesting to me.
IMHO the killer benefit of the Teensy is the Atmel microcontroller has the USB on the chip, whereas the Arduino uses a separate FTDI USB to serial chip. This means the Teensy is cheaper and smaller than an Arduino, but that's not the best part.
The Arduino needs the FTDI drivers installed, and always looks like an FTDI thing.
The Teensy runs the USB interface code itself, and you can program it to implement a range of USB devices. Dean Camera's LUFA (Lightweight USB Framework for AVRs) implements a range of USB devices (have a look, it's a long list) and is available under a nice MIT Open Source license.
The upshot of implementing USB on the Teensy is you can make almost any USB device you want, but more importantly, you can avoid the need for the FTDI driver. Your Teensy-based device can look like a keyboard, or a flash drive, or a serial device, and use the Operating Systems pre-installed USB device driver support. You could walk upto any reasonable computer and just plug it in.
Because the device can look like different USB devices, the talented Mr Camera's LUFA has several bootloaders. Some Operating Systems support DFU (Device Firmware Update), so you can bootload with nothing else. No avrdude, no driver, nothing. At worst it can be booted directly with an Open Source program called dfu-programmer, again without drivers.
The AT90USBKEY can also run LUFA.
It has a AT90USB1287, which Atmel says is:
128KB self-programming Flash Program Memory, 8KB SRAM, 4096-Byte EEPROM, USB 2.0 Low speed and Full Speed On-The-Go (OTG) Host/Device
for a similar price to an Arduino (in my experience Atmel do a low cost initial price to get interest, then push up the price later)[/edit]
If only Atmel made this stuff in through-hole DIL packages like PICs (are you reading this Atmel?). It would be near perfect; a breadboard friendly USB slave, able to implement many USB devices, with no need for device driver installation, and with DFU pre-installed, so no need for a ICSP programmer. This would let us breadboard a working USB machine for under £5.
Sorry, I got carried away. I bet you can guess what I want for my brithday? Yes a DIL package ATmega32U4!