I just replaced the tail lights on my trailer and the left one works perfect no problem, but the right one only the brake light and turn signal work but the running light won't work and I am stuck, any suggestions? Thanks in advance!!!

Light's need electricity to work,....check the socket to see if you have any

It should be getting power since the brake light and turn signal are working right?

Nope, the running lights use a different circuit, what side does the main wire harness run down the trailer?, I'm betting the left. 

If you were to read between the lines, they probably use Scotch locks on the brown wire going to the right side from the left, seen it a million or so times...Get rid of the Scotch Locks and used crimp on terminals.

Left turn/brake = yellow wire.

Right turn/brake = green wire.  ( here is a cheat for that one, the right navigation light on a vessel with lights is green).

Running lights = brown wire.

Ground = white wire.

GT

Thanks I will probably mess with it tomorrow and let you know what happens.

solder and shrink tube do it right do it only once

I never bothered with heat shrink and solder when it comes to freshwater crap, but salt is a different story, I have used the heat shrink and hot glue filled crimp-ons with great success.....   http://www.buyheatshrink.com/heatshrinktubing/heat-shrink-crimp-splices.htm

The thing about trailers is the wire quality, sort of like lipstick on a pig.

GT

Lipstick on a pig     GT where did you hang out in the early years LOL

You haven't met my dad...        He is a pretty fun dude to hang out with. 

GT