Apologies for my lack of knowledge, I am trying to teach myself how to work with electronics and have put together this circuit. Can somebody tell me why the relay is not working?
This doesn't work because the current thru the touch sensor is small and the gain of the transistor isn't large enough to amplify it to what the relay needs.
You haven't provided any specs on the relay, so I'll just make up numbers for sake of example. Let's say the relay needs 30 mA at 9 V to operate. You haven't provided a link to the datasheet of the transistor, so let's say it's guaranteed minimum gain is 50. That means the minimum base current to operate the relay would be (30 mA)/50 = 600 µA. It is quite unlikely that this touch sensor will let 600 µA pass.
Let's work backwards to see what the resistance between the touch sensor terminals would need to be. Figure 700 mV for the B-E drop of the transistor. That leaves 8.3 V across the touch sensor. By Ohm's law, (8.3 V)/(600 µA) = 13.8 kΩ. That's low for ordinary skin, but could probably be achieved by wetting the skin with salt water first.
However, beware of the current thru the body. If this touch sensor is passing current between different parts of the same finger, then you might only feel a tingle. If this touch sensor connects two fingers on opposite hands, then this would actually be quite dangerous. Nearly a milliamp flowing near the heart is a bad idea.
In any case, the solution is more gain. A second transistor could be used. That would also allow a resistor in series with the touch sensor to guarantee the current thru the body would be limited. This is something your circuit is NOT doing now.
Another way to build this circuit is with a MOSFET. It can be used in these applications because it requires almost no current to switch on and off. The gate of a MOSFET is like a capacitor; when it charges it allows current to flow. When you place your finger across the contacts, the gate charges and allows current to flow. Select the MOSFET for the current you need. The downside of this circuit is humans have the ability to collect static charge and this is bad for the MOSFET's gate as they are susceptible to ESD. It's a fun circuit to build however. You can change this by adding ESD diodes.
Some MOSFETs are so sensitive, you only need to wave your hand next to them; you don't even need to touch the contact.