"P" stands for "probability" and is usually given as the probability that what we found in the sample does not reflect the population from which the sample is drawn. So, if the P value is .10, there is an 10% probability, or chance, that what we found in the sample is not true for the population. We can be 90% certain that what we found in the sample is true of the population. If we were to draw repeated samples from the population, then 90% of the time we would expect to find a value at least as great as the one we found in the sample we are using.
The alpha level is the P-value that we as researchers decide to accept before we will be confident enough to release a finding. This is our predetermined acceptance level. The alpha level is not calculated, it is chosen by the researcher(s). In the social sciences, an alpha level of .05 is generally considered "acceptable." Many researchers will not accept a P value greater than .10. This means that only if the researcher is 90 (p-value = .10) to 95% (p-value = .05) sure of their findings will they submit the findings to a journal or release the information to a newspaper.
Although much emphasis is placed on "finding significance" at the .10 or .05 alpha level, it can be just as important to find no significance, when theoretically we expect to find one.
Though the academic world seems to regard a .05 or .10 level as one of the "sacred" markers of statistical significance, it really depends on the situation. In some cases, such as medical research being used to support release of a potentially dangerous new drug, we might want a more stringent level. On the other hand, "medical legal certainty" of a disability claim uses a less stringent definition of "reasonable probability," which is defined as 51% or greater certainty (Bennett, 1995).