a) There are two white #14-2 cables entering this box.  The #14-2 black wires are spliced together in one yellow wire nut, and the two #14-2 white wires are spliced together in another yellow wire nut.  No #14-2 wires are connected to either receptacle. This is as it should be.  

b) the receptacle on the right is a #12 ground-fault interruptor, the receptacle on the left is its mate, a plain modern (square) style outlet.  

c) The white wire which you found loose which you capped with a red wire nut is from a #12-2 yellow covered Romex cable, (and is thicker than the #14 white wires spliced together with the yellow wire nut).

d) The black wire which you found loose which you capped with a red wire nut is from a #12-2 yellow covered Romex cable, (and is thicker than the #14 black wires spliced together with the yellow wire nut).

e) If a,b,c,and d, above are correct then the end of the loose WHITE wire designated in c) should be bent in a semicircle and slipped under the unused screw on the WHITE inner side of the right hand side groundfault interruptor receptacle next to the existing attached WHITE wire, and

f) If a,b,c,and d, above are correct then the end of the loose BLACK wire designated in d) should be bent in a semicircle and slipped under the unused screw on the BLACK inner side of the left hand receptacle next to the existing attached BLACK wire.

g) I suggest that you turn the ciruit breaker off, while making the connections at e) and f).

h) I suggest that having tested the outlets and found them to be powered,
you again turn the ciruit breaker off, while inserting and fastening the
receptacles in the metal box.

i) Please feel free to telephone me at any time, but remember that I will not answer, because it takes me a long time to clamber out of bed. I will return your call as soon as I have managed to sit down at the table. 

j) Good night. I hope that you sleep well.