My mom’s sister’s son is my first cousin, and our last common ancestor is our grandma. My grandma’s sister’s grandson is my second cousin, and our last common ancestor is our great-grandma. My great-grandma’s sister’s great-grandson is my third cousin, and our last common ancestor is our great-great-grandma. So the number is how many generations back your last common ancestor is. The little trick is that the number is how many G’s are in the word for your last common ancestor- so your fifth cousin and you have a great-great-great-great-grandparent in common (five G’s).
Removed refers to a difference in generation. So my first cousin’s son is still my first cousin, but once removed. That kid’s son is my first cousin twice removed. It starts with whoever’s closer to the last common ancestor and then you count down from that level.
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