Oxygen Radicals are a form of free radical, a type of atom, molecule, or ion which can be extremely chemically reactive (Wikipedia). Simply put, this means that these molecules (for example) can easily react with other molecules, creating a chemical reaction that can be benign or destructive to both the molecules in question, and to the surrounding chemical or biological medium.
Since oxygen is a corrosive, oxygen radicals can easily bind with tissue molecules and cause damage. Tissue damage from oxygen radical activity has been observed and indicated in the spread of cancer tumors, and they have also been linked to DNA damage, which can cause or encourage cancer. Studies are ongoing, but it seems clear that the cellular damage caused by oxygen radicals over time, assuming the cells don't die, can encourage cancer by linking into the DNA instructions to continue uninhibited growth; damaged cells can no longer grow according to their natural inclinations, and damaged DNA sends out the wrong signals, causing too much cellular reproduction.