Both Teflon and Velcro are trademarks for artificial substances. Teflon is a coating used because it is low friction. Velcro is used because it sticks very well to other Velcro; it is used for simple shoes. A Velcro strap attaches to another Velcro strap without the need for a buckle. As a side effect, other substances tend to stick to Velcro in unwanted ways; this is particularly problematic for cotton (a common part of socks).
When used idiomatically, for example, applying to American presidents, "a Teflon president" is a president to whom no scandal will stick. "A Velcro president" is a president to whom all scandals seem to stick, even those that ought not negatively affect him personally.
Originally, "Teflon" was used as an idiomatic adjective with wide usage (such as with John Gotti, "The Teflon Don," a noted American Mafia leader who for many years escaped punishment, being found innocent in several criminal trials). Since Velcro sticks to anything, journalists have adopted "Velcro" as the opposite of Teflon, and idiomatically use the word to spice up news stories.