The fleas that take over our households are almost always cat fleas otherwise known as Ctenocephalides felis. Cat fleas are just as likely to feed on your dog as they are on your cat. And these types of fleas can bite people occasionally.
Adult fleas are built for hiding in fur because their unusual flattened bodies help them circumnavigate between pieces of fur or hair. This is thanks to their backwards-facing spines on their bodies that help them cling to fur when your pet is moving. Contrast this to our relatively hairless bodies which don't make great hiding places for fleas, and it's much harder for them to hang on to our skin.
When an invasion occurs, these bloodthirsty fleas are competing for your pet, and may begin biting your body instead. These types of bites tend to occur on the ankles or lower legs and they do itch, especially if you're allergic to them.
Do be wary though as while fleas rarely tend to house themselves on human skin, they can and will live happily in a human home regardless of a pet presence and if fleas do decide to settle into your home then take the necessary precautions.