Multi-National Corps Iraq (MNCI, as seen on the left of this patch) was the intermediate level command, stuck between the Division commands (two star generals) and the Multi-National Forces Iraq command (the 4-star general in charge of all Iraq, once GEN Petraeus). In reality, they often acted as a go-between for the higher and lower HQs. So you can image there was some animosity between MNCI and MNFI, especially since their offices were literally next door to each other.
The "OIF Infinity" on the right of the patch referencing the military's system of numbering OIF rotations. It started with OIF I in 2003, but within a few years the system of sequential numbers became too confusing, so they went with simply using the years which the rotation spanned (i.e. OIF 06-07). Infinity is a clear reference to our seemingly open ended commitment.
"JOC" stands for Joint Operations Center, same as a Tactical Operations Center but in a joint environment. So a "JOC Rat" would would be the same as a "TOC Rat", which is a slang term for someone who works in a TOC all day and rarely sees the light of day. Similar species to a TOCroach. Yes, this patch says Rates instead of rats, but misspellings such as this are common and most likely why the patch was being sold to just anyone and not the person who ordered it.
The "spear" the rat is holding is taken strait from the middle of the MNCI shoulder patch (shown left), and the reference to "here kitty kitty" is about the "lion of babylon" that is the major item on the MNFI shoulder patch (shown right).
