Definition of Per Stirpes
"Per stirpes" is a latin term meaning "by roots" or "by stocks". It is normally used with the word "issue", meaning children, grandchildren and other lineal descendants. In fact, the phrase "issue per stirpes" is one of the only "old-fashioned" legal terms that I still use, as it is the quickest way to describe a particular form of division of property among successive generations.
You will be familiar with the typical picture of a family tree, where "You" are the trunk and your parents, grandparents etc. are the branches. Now consider the roots of the family tree, which represent your issue (children, grandchildren etc.). In this diagram, "You" have three children, three grandchildren, and six great-grandchildren:
If you give the residue of your estate to "my children alive at my death in equal shares", and Child 1 were to die before you, then the share of Child 1 would be be divided between Child 2 and Child 3, and the children of Child 1 (i.e., Grandchild 1) would not receive anything.
On the other hand, if you give the residue of your estate to "my issue alive at my death in equal shares per stirpes", and Child 1 dies before you but Grandchild 1 is still alive at your death, then the "root" represented by Child 1 still exists. Instead of Child 1's 1/3 share passing to the other "roots", it will trickle down to the next living representative in Child 1's "root" (i.e., Grandchild 1). On the other hand, if Child 2 dies before you, then Child 2's "root" and share are completely eliminated because Child 2 doesn't have any children, and the residue of your estate will get divided 1/2 to Child 1 (or other issue in Child 1's "root") and 1/2 to Child 3 (or other issue in Child 3's "root").
Let's look at one more example. Suppose Child 1 and Child 2 are alive at your death, but unfortunately Child 3 and Grandchild 2 died in a plane crash together shortly before your death. The residue of your estate will be divided as follows: