Where is the turn signal relay for my 2008 Taurus X?

You have a light out.

Did you put a aftermarket radio in ? The clicking sound comes out the driver side speaker . It's generated by the BCM body control module .
Turn Signal Lamps
The STOP/TURN fuse in the instrument panel (I/P) fuse block supplies battery positive voltage to the body control module (BCM) for the turn signal lamps. When the turn switch is placed in the left or right position, ground is applied to the BCM on the left or right turn signal switch signal circuit. The BCM then applies voltage to the appropriate front and rear supply voltage circuits.
Grounds for the various lamps are as follows:
• G100 supplies the right front lamps
• G101 supplies the left front lamps
• G400 supplies the right rear lamps
• G401 supplies the left rear lamps

Your best bet would be to take it to a qualified repair shop before you start replacing parts , like the BCM ,which would need to be programmed at the dealer or other thing's will stop working too .

flasher/blinker is busted. this is located under your dash board. remove steering column cover, turn on LEFT turn signal (yes, the one that is working), then you will hear a clicking sound. locate the relay that causes the sound. there you can see 2 relays (one for LEFT, other one for RIGHT turn),replace busted one.

the signal indicator lights are suppose to flash in perticular time interval which is connected to a timer relay when the light flashes faster that means the other light is not working and the relay is getting double the current and moving faster so check the other light either the bulb is fuse or the current is not reaching up to the bulb

you might have a blown bulb. turn on 4 way flashers (hazard lamps), walk around vehicle and see which one is not working. Replace burned lamp.

ok thats simple for for a long time has been known for bad grounds also check where main wire plugs into fire wall see if it is loose but most likly its a bad ground in signal

Normally with the blinker blinking really fast, it's telling you that one of the bulb is blown, front or rear signals. Check your fuse, and fuse relay to see if it would fix the promblem

Replace the turn signal switch. Bad contact inside causing a crossing of the circuits. When the turn signal is applied it should open the brake light circuit allowing for just the signal to work. Doesn't sound like the switch is doing it's job.

the next step and almost certainly the problem is the turn signal switch, have seen this symptom on other years around yours and switch fixed it.

Sounds like your front turn bulb burnt out. When the turn signal operates fewer bulbs it works faster. You need to change the right front bulb.