Six Sigma has its roots in the manufacturing sector, and a primary focus of the Six Sigma methodology is process and quality improvement via reduction of process variation. The term "sigma" is a concept from the statistics field that is a measure of the standard deviation of a set of data. The term "six sigma" reflects the ultimate goal of having a process that has only about three defects (failures to meet customer and manufacturing specs) out of a million opportunities. Such a process has a sigma level of 6.0. This level of performance is usually not realistic in service settings, but the name is used nonetheless. Most Six Sigma projects follow the DMAIC methodology to reduce variation and improve performance to meet customer expectations.
To understand the impact of variation on performance, consider the two graphs below. Suppose a pizza company is tracking their delivery times relative to the time that they tell customers to expect their pizza. Certainly customers do not want their pizza to arrive late. But there may even be customers who do not want it to arrive early, for instance if they are placing the order when leaving work to go home, or if they plan to take a quick shower first. The company has decided based on customer feedback that they want to be sure to arrive within 8 minutes of the planned arrival time at a customer's house.
In the first graph, you see that this company is usually arriving within the window they consider acceptable, as shown by the large portion of the curve that is within the two dotted lines indicating the lower and upper limits. However a percentage of customers are receiving their pizza either more than 8 minutes before or more than 8 minutes after the time they were told to expect delivery.

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Often to change process performance, a group will try to shift the average, for instance in this case the company might try to just achieve earlier delivery in all cases. But this won't please customers who don't want their pizza too quickly, and there will still be enough variation in the process that if something changes and the average increases again, performance will suffer.
Only by reducing variation will performance be substantially and sustainably improved. In the next graph, you can see the result of an effort to tighten up the pizza delivery process, so that all customers receive their pizza within 8 minutes of the expected delivery time. Furthermore, even if there is a short-term problem or a long-lasting change, the delivery time could increase or decrease somewhat without resulting in increased defects.

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Note that this improvement has occurred without even shifting the average time, or the upper and lower spec limits.