Alpha (also known as the "alpha coefficient") is a way of analysing the value that an active fund manager adds to his or her fund. If you invest in an active fund, then you want it to beat the market rather than simply track its performance (or worse still, underperform the market).
However, even if the manager manages to beat the market, that doesn't mean that they have added "alpha" (ie demonstrated skill). After all, a manager could potentially beat the market by putting all of their fund's money into a single stock that then happens to do very well, but that would involve taking a huge amount of risk, and none of their investors would be terribly happy about it.
So alpha measures the rate of return made by a portfolio relative to the return on its benchmark, but only after adjusting for investment risk (which is measured by beta).
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For example, suppose a fund achieves a 30% return while its benchmark rises by 8%. But say the fund is three times more volatile than the benchmark (in other words, it has a beta of three). That would make the fund's alpha 6% (calculated by multiplying the benchmark return (8%) by the fund's beta (3) so in this case, 24% then subtracting from the fund's return). This means the fund has returned 6% more than you would expect, once an allowance has been made for the additional risk that the portfolio manager has taken on.
See Tim Bennett's video tutorial: What is 'alpha'?
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