Surface tension occurs because water molecules attract on another. That means that water prefers to form compact shapes with little surface area. Creating a large, extended area, as you do in a soap bubble, is actually opposed by surface tension. This is why you can't get stable bubbles with pure water: the bubble wants to collapse into a compact shape.
Soap changes the situation. Soap molecules have one end that is attracted to water molecules (the hydrophilic end); the other is not (the hydrophobic end). In water, the soap molecules will try to arrange themselves so their hydrophilic ends are in the bulk of the water but the hydrophobic ends are at the surface of the water (or clustered together to minimize contact with the water - these clusters are called micelles). This means it is no longer so costly for the water to form large surfaces, because the hydrophobic soap ends are the parts that are exposed at the surface instead of the water itself. In a very soapy solution, it's easy to form bubbles: you will have two layers of soap molecules, one on the inside and one on the outside. In between, a layer of self-attracting water molecules will hold the bubble together.