I have a Canon EOS T3i 600D. When I use a Canon-branded lens (18-55mm), the camera works well. When I change to the Sigma macro 18-250mm and turn the camera on, I can hear a ringing noise, like ringing of the ear type noise. What's causing this?
If the lens is similar to the Sigma 18-250mm f/3.5-6.3 DC Macro OS HSM, it uses an ultrasonic (Hypersonic, in marketing parlance) motor for focusing. These motors can be smoother (less backlash or overshoot) than traditional electromagnet-based rotary motors, but apparently the "ultrasonic" ratcheting is not quite beyond your hearing range.
This is nothing to be concerned about if the sound is not bothersome to you. It might produce some interesting reactions in pet photography, though.
I believe a lot of cheaper hypersonic montors (HSM) [or USM for Canon glass] don't have great quality control and can produce that kind of noise. Likewise, I've personally had a Canon USM lens that was vibrated too much in a case emit that kind of noise, too. Your best bet would be to send the Sigma back for warranty service (or take it into a shop) to get it checked out.
As someone with sensitive ears, I can occasionally hear the whine of the motor when it's desperately trying to focus on a subject in motion, but in general, noises you don't expect right after you turn the camera on implies some kind of malfunction. Best of luck to you!
tl;dr: I think it is the image stabilization.
Additional info: Check whether you can hear the noise on all stabilized lenses (ask a friend or check at a camera store if there is no other choice). I certainly hear the stabilizer every time it's operating on my two stabilized Nikon lenses and some more of my friends. Reading the other answers, apparently the sound is out of the hearing range for some people (or probably the most?). On Nikon lenses the stabilizer is coupled to the lightmeter, - the stabilizer is switched on and off with the lightmeter (the absolute time can be set in the menu). I am not familiar with the Canon system, but 'the internet' reports a similar feature for canon (in a different context, though):
You may also be able to hear the click. By checking if you can hear the noise on every IS lens you can find out whether the noise is in the normal range.
When I change to the Sigma macro 18-250mm and turn the camera on, I can hear a ringing noise, like ringing of the ear type noise. What's causing this?
All moving parts are located in objective: the aperture iris, the IS, the focusing motor. Whatever is broken is located in objective. And, if you hear the sound right after turning the camera on, when no moving parts are expected to move, it is natural to suggest that objective is malfunctioning in some way (you may make it more clear if you describe how well the objective works).