My question is what is this "plug" here- Some times my headlights do not work, but then if I sqeeze or move around this wires, my lights come on. What is the deal here?

 

find some old pins and rewire it!dont forget to solder the wires.

 

There is also a lot of grey looking grease in there. I don't think grease should be in electrical wiring, or am I wrong?

 

<TABLE WIDTH="90%" CELLSPACING=0 CELLPADDING=0 ALIGN=CENTER><TR><TD>Quote, originally posted by TapOrSnap &raquo;</TD></TR><TR><TD CLASS="quote">There is also a lot of grey looking grease in there. I don't think grease should be in electrical wiring, or am I wrong?</TD></TR></TABLE>

If it's di-electric? grease, then it should be there (to prevent corrosion). Regular grease; no, corrosion will still form.

Yeah, definitely de-pin/resolder your wires -- Junkyard diving time.