"2nd cousins" means a child of
your parent's cousin
if your second cousin and you share the
same great grandparents. But your second cousin and you
do not share the same grandparents. Also, your parent's cousin is your first cousin, once removed.
"
3rd cousins" means a child of
your parent's second cousin, once removed.
Only if your third cousin and you share the
same 2nd great grandparents, and
do not share the same great-grandparents
and grandparents. It could have another same meaning if it is a child of your grandparent's cousin's grandchild.
"4th cousins" means a child of your parent's third cousin, once removed.
Only if you share the same 3rd great grandparents.
"5th Cousins and etc." means the same meaning with different generations by different grandparents in a number of genertions.
Why once removed (or depends on a number of generation)?
Those words, "once removed", is used to describe a relationship that two people are from different generations. For an example, your mother's cousin is your first cousin, once removed. This is because your mother's cousin is one generation younger than your grandparents and you are two generations younger than your grandparents. This is one-generation difference equals "once removed."
Another meaning . . .
"Twice removed" means there is a two-generations difference. You are two generations
younger than a first cousin of your grandparents, so your grandmother's cousin and you
are first cousins, twice removed.
Hope that help.
PS - You can visit
this. (EDIT!: I'll draw a map of family tree with describe the meaning of relationships. So it will take a while to be done.)