I know that it is a very basic question but I really can not find an answer to it.
I would like to set my op amp to inverting amplifier mode so, I made the following circuit using LM358.

I vary the voltage of the inverting input of the op-amp using a variable resistance and a voltage divider.
I put 1 volt as an input, I expected to have -1 volt as an output but that doesn't happen. The output voltage is always zero ( ground ) or a very small value. I changed the input voltage from 2 to 12 volt, but the output voltage remains zero.
I used another power supply (5 volts) put I've got the same results which is zero volt at the output.
I saw some circuits on the internet and I found that the op amp is supplied with +9 volts and -9 volts, and the non inverting input is connected to ground. Does that mean I have to use a 3-terminal power supply ? Is there any other solution ?

 

You're using the LM358 with a single supply. Output voltage can only get very close to ground. If you want output to swing negative, you need to use split supplies.

 

I expected to have -1 volt as an output

An op amp cannot output a voltage outside the range of its own power supply, and usually they cannot get close to the power rail voltages unless they are specifically designed for that. There are "rail-to-rail" pampas and there are op-amps that can only approach one of the two power rails but not the other.

Also, op-amps cannot sense input voltages outside the power supply voltages. Read the common mode range specifications.

 

If you input -1V to your circuit then the output will be +1V, as would be expected.

If you want to amplify a plus and minus signal with a single supply then you need to bias the op amp plus (+) input at 1/2 the supply voltage with a resistive divider (say two 1kΩ resistors) and use series capacitors to couple the signal at the input and output.
The capacitors must be large enough to carry the lowest signal frequency for the input and output load impedance you have.
Of course that technique will only work for AC signals, not DC.