Simple LED circuit is not working
What connects the resistor to the LED?



Absolutely nothing.

It's hard to work out which holes the LED is plugged into, but I think your current circuit looks something like this:



You need to connect that resistor and LED together.

For your edit:

Now your circuit looks like this:



Behind the 5 pins in that column with the ground wire there is a big chunk of metal connecting them all together.  It's no different to if you'd taken the LED and twisted the pins together into one.  There's no way the power can get up one and down the other when they are connected together like that.  You need to separate them into different columns.

And this is how the (black) connections inside are laid out, and how your (red) electricity is flowing:



And here is how the circuit should look, using a program called Fritzing.  It's a pretty basic design tool, but great for showing wiring.



You can see how the green highlighting shows the columns that are connected.