"First cousins" share the same grandparents but not the same parents; that is, your first cousins are the children of your aunts and uncles. "Second cousins" share the same great-grandparents...that is, they are the same-generation children of first cousins. "Third cousins" share the same great-great-grandparents. For each further stage, add another �great'.
If one of your first cousins has a child, then that child is your "first cousin once removed". The 'once removed' element just means that a further generation is involved. In the same way - looking in the opposite age-direction - you are the first cousin once removed of any first cousin of your parents. All that �once removed' means, then, is that the two people are a single generation apart. Your grannie's first cousin is your first cousin twice removed, because there is now a two-generation gap. And so it goes on.