The To-Complete-Performance-Index (TCPI) allows a projection of the anticipated performance required to achieve a goal.
As a simple example: You are driving in your car to a friends house. You promised that you would arrive at 3pm. It is now 2:15pm and you have 30 miles to go. Your TCPI is the speed that you need to drive in order to arrive on time. (This is obviously not a perfect example for the TCPI, but it gets the point across: The TCPI defines the performance required in order to achieve a previously set goal.)
The PMBOK Guide Fourth Edition defines TCPI as the calculated projection of cost performance that must be achieved on the remaining work to meet a specified management goal, such as the budget at completion (BAC) or the estimate at completion (EAC). That is is why there are two formulas – one calculates the TCPI to achieve the BAC and one to achieve the EAC.
TCPI can also be compared with the Cost Performance Index (CPI). This can provide additional performance information. For example, if the TCPI is greater than the current CPI then future efficiency must improve if the project is to achieve the BAC or EAC.
If the comulative CPI falls below the baseline plan, all future work of the project will need to immediately be performed in the range of the TCPI (BAC) to stay within the authorized BAC. Whether this level of performance is achievable is a judgement call based on a number of considerations, including risks, schedule, and technical performance. Once management acknowledges that the BAC is no longer attainable, the project manager will prepare a new estimate at completion (EAC) for the work, and once approved, the project will work to the new EAC value.
Review the complete definition plus examples from the PMBOK Guide Fourth Edition starting at chapter 7.3.3