The Teensy is a complete USB-based microcontoller development system, in a very small footprint, capable of implementing many types of projects. All programming is done via the USB port. No special programmer is needed, only a standard "Mini-B" USB cable and a PC or Macintosh with a USB port.
Teensyduino is a software add-on for the Arduino IDE. It adds the ability to build and run sketches on the Teensy and Teensy++!