You have a Falcon or Jasper Console

I lump Falcons and Jaspers together because for modding purposes they are the same. If you would like to know speficially whiche one you have I have provided a labeled image of the power sockets here:

solid top bar is a falcon, split top bar is a jasper

Both Falcons and Jaspers can be easily modded on even the very current most dashboard and get great boot results using the new-ish (released 2015) RGH 1.2 method. I have a really detailed tutorial for RGH 1.2 here.

For those that are interested in the basic tools that would be used to RGH a Falcon or Jasper console you would need two primary items:
-Some way to read and write to the nand. This could be a JR Programmer (v1 or v2), a Nand-X, a Super Nand Flasher, or several other options
-A glitch chip such as a CoolRunner Rev C or a Matrix V1.