In simple terms, TTL is a method of measuring the amount of light (ambient or flash) coming from a scene by placing sensors behind the lens. TTL metering can be used to control both the camera and the flash.
For example, when a camera is used in full-auto mode (also called Program mode), it uses TTL metering to determine both the shutter speed and aperture for what it thinks is the correct exposure. Or when used in full-manual mode the camera uses TTL metering to indicate (usually with a + or - exposure value) whether the exposure is going to be correct for the shutter speed and exposure you have chosen.
For TTL flash or strobe photography the camera uses TTL metering to control the strobe itself and quenches the strobe when it thinks the scene has been lit properly - so in a way the camera is 'talking' or 'communicating' with the strobe. For this to work the flash must understand the TTL protocol or language that the camera uses. Unfortunately different manufacturers use different protocols, making it difficult to swap strobes between manufacturers.