"Heirloom" should be defined as an exceptional plant that had been handed down from generation to generation.  Among cloned woody plants (trees, shrubs, or vines) this is  easy to verify.  Heirloom trees, shrubs and vines are easy to compare and acknowledge their identity.  Among seed grown plants (annual flowers and vegetables) it only means that the particular variety is a "pure breed" rather than a "mutt".  There original plants are no longer present to compare with.  Most plants have 2 sets of chromosomes.  In a "pure breed" the 2 sets should be essentially identical.  Any offspring between heirlooms should turn out the same.  You really can't really prove that a seed grown plant has actually been handed down for generations.            

"Hybrids" plants are "Mutt"s.  A hybrid tomato is the result of taking the pollen from one heirloom tomato and fertilizing the flower of a different heirloom tomato.  The resulting seed contains a plant with 2 different sets of genes.  This often creates "hybrid vigor", especially if the parents each had exceptional and distinct characteristics.  Every time more seeds are needed, the supplier must have both varieties present and make certain that the pollen from one is the only pollen available to fertilize the flowers of the other.  (This is quite labor intensive.)  One variety is usually chosen as the "male" and the other as the "female".  The resulting seeds produce 'hybrid' plants with mixed genes.  The seeds created by this hybrid plant will produce offspring with many different characteristics.  Some will resemble each grandparent and others will resemble the hybrid. 

Hybrids can be turned into Heirlooms.  (Or conversely, all heirlooms were once hybrids.)  The breeder discards all offspring that don't resemble the hybrid.  The seeds of this next generation are planted and the process repeated.  With each generation the genes become more "stable" with fewer variations.  With persistence, the mutt can be turned into a pure breed within 10 generations.