The software used here is Adobe Illustrator
This technique is not only meant for circular rings, you can also use it for text and other forms to provide the illusion of dimension.
1.) First of all, you need to set the fill to none and the stroke to any colour. Here, I am using the red colour.
2.) Then you need to choose the Ellipse tool and draw a circle by holing the Shift key while dragging to obtain the perfect shape.
3.) Change the stroke weight in the stroke palette to create a thicker outline.
4.) To convert the outline to the form of an object navigate to Object > Path > Outline Path.
5.) Create a copy of the circle by using the selection tool, Alt drag (Mac Users: Option drag). Repeat the process to create another copy.
6.) You need to change the fill of each of the objects to get a different colour.
7.) Now choose all five rings. From the align palette select Vertical Align Centre and then click on horizontal Distribute Centre. Then select the second and fourth
8.) Without removing the selection, navigate to the pathfinder palette and choose divide.
9.) To ungroup the objects, select Object > Ungroup
10.) Select the portion of the rings where you want them to overlap. Then fill the portion of overlapping with the colour similar to the overlapping ring. To match the colour, use the eyedropper tool.
Let me show you how to do it. First of all, select the portion. Check the colour swatch to see whether the fill is active. Then using the eyedropper click on the colour you want for the fill. You have to do this for the middle ring and the right ring.
11.) Now the rings seem to overlap with each other. But still the rings appear to be split into many sections. Choose all the sections of each ring and navigate to Object > Make compound path to join the rings together. You can do this with ease by selecting one section of the ring and then choosing Edit > Select > Same fill colour. Then pick Object > Make compound path.
12.) At the final stage, you can select all the interlocking rings to group them.