What is the effect of surface tension on an air bubble inside water?

Surface tension happens at the boundary of a liquid and something else. In your case this something else is either the air of the bubble or the bottom of the container.

Look at bubbles form in a glass filled with a carbonated drink. Some form on the walls and on the bottom. They stick there until they get big enough so that the buyant force pulls them off.

Surface tension is rightly named: it is a tension in the surface that separates a liquid and another fluid, air in your case. This tension is at the origin of forces: e.g, if the surface is curved, the force tends to straighten it (as when you exert a tension on a piece of string: it will straighten). In a small spherical bubble which is far from the walls of the container and from the surface, surface tension acts all over the surface, exerting a force toward the center of the bubble: this force is resisted by an opposing pressure in the bubble. This is known as Laplace law.

If the bubble is at the bottom, then surface tension is also at the origin of a force at the line of contact of the water-air interface and the bottom of the container. This force is proportional to the value of the surface tension (a constant that only depends on the fluids you have and temperature) times the length of that line.

If the bubble is small enough, this force can resist the buoyancy force. Imagine now that you inflate the bubble, the volume and thereofre buoyancy will grow faster than the perimeter of the contact, and at some point the buoyancy force will be able to detach the bubble.

(Note that in order to have a complete picture, one should also consider the shape evolution of the bubble, which is governed by other surface energies: the solid-water energy and solid-gas energy, as the angle of contact governs the proportionality mentioned above)