What relative name would be used to describe my father's cousin's child, my cousin's child -and- what's a first cousin once removed, a second cousin once removed?

Your father's cousin's child is your second cousin. And your father's cousin is your first cousin once removed.

If you are referring to your father's first cousin, then the child of that first cousin would be your second cousin. But from your point of view, your father's first cousin is your own first cousin once removed.

This kind of thing has always driven me crazy, but I found a chart that makes it easier to figure out. Your second cousin is the child of your first cousin once removed. Your first cousin once removed is the first cousin of one of your parents. Get it? You may have to read it more than once to get your brain wrapped around it. I always do! Anyway, you can find the chart at:

http://www.genealogy.com/genealogy/16_cousn.html
That makes it much easier, more so if you write on paper. That's what I did.

Imagine you and I are first cousins (i.e. either your mother or father was the sister or brother of one of my parents.) Your children would be my first cousins once removed (removed one generation.) Likewise, my children would be your first cousins once removed. Our children would be second cousins to each other.

With all due respect to the compilers of the chart to which BB has drawn attention I must point out that this only gives part of the picture.

The children of your first cousin are first cousins once removed and their children are first cousins twice removed and so on.

Second cousins are the children of your parents' first cousins. Their parents and their children are also second cousins but (and this is the point) once removed upwards in the first case and downwards in the second. From the standpoint of your cousins removed downwards, you are a cousin removed upwards.

Third cousins are the children of your parents' second cousins, and so on until the proverbial 42nd cousins are reached.

One rule to bear in mind is that your plain first, second, third, etc. cousin (ie without any removes) must be of your generation.

If you can follow all this, then you will readily understand that from the point of view of your third cousin once removed upwards, you are that person's second cousin once removed downwards.

The best (and the clearest) explanation of family relationships I have ever come across is to be found in 'The Dunlop Book of Facts' which was published in the UK in 1964 and was a precursor of the 'Guinness Book of Records.' If you can locate a copy it will be well worth the effort.

I'm somewhat thicker than most, but the following site explains it very clearly.

Cousin (a.k.a "first cousin"): Your first cousins are the people in your family who have two of the same grandparents as you. In other words, they are the children of your aunts and uncles.
Second Cousin: Your second cousins are the people in your family who have the same great-grandparents as you., but not the same grandparents.
Third, Fourth, and Fifth Cousins: Your third cousins have the same great-great-grandparents, fourth cousins have the same great-great-great-grandparents, and so on.
Removed: When the word "removed" is used to describe a relationship, it indicates that the two people are from different generations. You and your first cousins are in the same generation (two generations younger than your grandparents), so the word "removed" is not used to describe your relationship.
Once Removed: The words "once removed" mean that there is a difference of one generation. For example, your mother's first cousin is your first cousin, once removed. This is because your mother's first cousin is one generation younger than your grandparents and you are two generations younger than your grandparents. This one-generation difference equals "once removed." Twice removed means that there is a two-generation difference. You are two generations younger than a first cousin of your grandmother, so you and your grandmother's first cousin are first cousins, twice removed.

http://www.genealogy.com/genealogy/16_cousn.html

Your aunts' and uncles' children are your first cousins. Their children are your first cousins once removed ("removed" one generation.) Your children and your first cousin's children are second cousins. Your grandchildren and your cousin's grandchildren are third cousins. In addition, your mother's first cousin is your first cousin once removed.

Second cousins have the same great-grandparents, but not the same grandparents.

http://www.searchforancestors.com/utility/cousincalculator.html

John and Michael are brothers.

John has a son called Joe.
Michael has a son called Mark.

Joe and Mark are first cousins.

Joe has a son called Jim.
Mark has a son called Mungo.

Jim and Mungo are second cousins.
Jim and Mark are first cousins once removed.
Joe and Mungo are first cousins once removed.