I've seen a lot of confusion out there on what types of razors are OK to take with you in your carry-on baggage. This post is just a quick and basic attempt to clear up any confusion. Let me just start by saying that electric razors are OK, but straight razors are a no-go.
These two photos below show a safety razor and the removable blade that they use. They're making a come back thanks to the close shave they provide and the extremely affordable price of the replacement blades. It basically unscrews and you can insert a razor blade. Because these razor blades are so easy to take out, safety razors are not permitted in your carry-on luggage with the blade. The blades must be stored in your checked luggage.
Photo Courtesy of Charlie Esser @ Flickr
Photo Courtesy of Scottfeldstein @ Flickr
Photo Courtesy of Cisc1970 @ Flickr
This is a disposable razor. They come in two types. The kind that is completely disposable (handle and all), or the kind where you replace them with cartridges. These are permissible in carry-on luggage with the blade and replacement cartridges.
Photo Courtesy of Dotbenjamin @ Flickr
This gentleman is shaving with a disposable razor. Good to go!
Photo Courtesy of Noob @ Flickr
This gentleman is shaving with a safety razor. No-go! (OK without blade)
Photo Courtesy of Emoneytg @ Flickr
This gentleman is shaving with a Samurai sword. No-go! (Swords must be checked with luggage)