What is the role of auxin in plant phototropism?

Starting with the basics: 
1. Auxin is a plant hormone, so it is a chemical substance involved in regulating the action of certain cells within an organism.
2. Phototropism is the process by which plant grow in response to light. Tropisms are responses to directional external stimuli and here Photo- indicates it is a response to light. 

Now, back to the actual question: 
Auxin increases the flexibility of plant cell walls which are the main factor limiting plant cell growth. As a consequence, plant cells tend to grow faster in the presence of auxin. When a plant senses a directional light stimulus it redistributesits auxin molecules to the side that is further away from the light. As a consequence, the plant cells that are further away from the light experience a greater auxin concentration and grow faster, causing the shoot to bend towards the light. This mechanism allows plants to optimize the amount of light they receive in order to increase photosynthesis rates. 
The figure below summarizes this quite well (note that IAA is simply the type of auxin that regulates phototropism in plants)



Note: Remember that it is NOT the difference in the level of production of auxin amongst plant cells that explains phototropism but the REDISTRIBUTION of existing auxin molecules.