Performance attribution explains the economic source of returns.
This can be done in a number of ways. For instance, one can take the investment process as a given and then explain the returns by the extent of a fund's exposure to shared (uncontrollable) market factors (factor attribution) or explain active returns by (uncontrollable) return differentiation (market attribution). (If all market returns were the same, then all portfolios would match their benchmarks. So any non-zero active return can be completely attributed to return differentiation.) These market approaches to performance attribution are intended to inform a portfolio manager about how the market moves not controlled by him or her did (and thus, may again in the future) affect a fund.
Alternately, one can take the market process as given and then explain active returns in terms of the investment decisions employed (decision attribution). If a fund perfectly matched its benchmark then, no matter how the market evolved, the active return would be zero. Thus, all non-zero active returns can be completely attributed to the controllable steps in the decision process that caused the fund to be constructed differently than the benchmark. This is intended to inform a portfolio manager about how the investment decisions he or she controlled created the active return and indicate what types of decisions he or she was good at and, perhaps, indicate what he or she might again be good at in the future.
Also, returns are often simply formally decomposed, in one way or another, into terms and are then (formally) said to be “explained” by the terms of whatever formal decomposition is offered (possibly a mix of controllable and uncontrollable factors). This approach then needs to argue that there is economic sense to the results despite it being difficult to find a precise economic question that it addresses.
It is a quirk of these models that in the very simplest of cases, (arithmetic, single period, single currency, buy-and-hold, two-step, fund-level attribution) all of these approaches can be modeled by the same formalism, that of Brinson attribution:
Allocation = the sum over all categories of the active category weight times the category return of the benchmark.
Selection = the sum over all categories of the fund's category weight times the active category return.
Here, the categorization employed is specified by the specific explanatory intent of the attribution type. For instance, for decision attribution, if the first decision allocates to sectors, then the category is sectors.
Finally, for applicability within modern market theory, one can also address risk and risk adjusted returns (risk (-adjusted) attribution) in ways similar to each of the above approaches.

No numerical example is supplied since, even in the simplest case, it is easier to understand the concepts involved then to gain any insight from a numerical example.