Is there a basic definition of Significant result and Non-significant result? Is there a basic or widely accepted definition of Significant Result and Non-Significant Result?  Seems that every where I look they want to get into problem solving without defining what those terms actually mean.
 A: Unfortunately, in particular situations one may be forced to try to infer meaning from the context in which such a phrase appears. "Result" in particular can be ambiguous, but in some cases even what "significant" actually means may be a little hazy.
"Significant" implies the use of significance testing, which strongly suggests that there was some Neyman-Pearson style* hypothesis test performed and that the null was rejected at some given significance level (or conversely, not rejected when the term 'non-significant result' was used). 
* it's possible some more nearly Fisherian approach was used, but the particular form of the phrasing would make me think Neyman Pearson.
Often, however, the same people that tend to use terms like "significant result" often tend to gloss over formalities like explicitly stating a null and alternative or even mentioning their significance level, so while there's an implication of a formal hypothesis test underlying such a statement, one must look to the context to try to figure out what it actually means (What null, exactly? What alternative? What test statistic? What assumptions? What significance level?) in a specific instance.
