Significance is an English word that has been hijacked by statisticians. In general usage the term significant difference means an important difference, but "significant" in this sense is subjective. On the other hand, statistical significance is objective, and is based on the concept "p0.05". In an experiment, a difference is detected by challenging a null hypothesis of "no difference". When p is less than 0.05, the null hypothesis is rejected - when p is greater than 0.05, the null hypothesis is accepted. So what does p0.05 really mean? A p value of less than 0.05 means that there is even less than a 0.05 probability of getting the observed test statistic if the null hypothesis is true. This is an incredibly misunderstood concept, by even experienced users of statistics. See also Probability.
Results of a clinical experiment are said to be "statistically" significant when they are highly unlikely to be attributable to "chance" and arguably more likely due to "psi". The chance probability is reported as 'the significance level'. Generally, to be considered significant, the chance probability must be less than 1 in 20 (5% or 0.05).