Walking is a form of ground-based movement slower than dashing. It is performed by slowly tilting forward on the control stick or d-pad while on the ground. In smash games that can be played without tap jump it's possible to walk by tilting the control stick diagonally up in the direction you want to walk to. While it is slower than dashing (though by exactly how much is dependent on the character), characters retain the use of their tilts and smash attacks while walking, giving it technical use. Characters have three different walking speeds depending on how far the control stick is tilted, with the slowest of these animations often being a sort of tip-toeing. In Brawl, however, if a character is controlled with a Wii Remote, they will always walk at their max speed. Hitting the attack button while walking will result in a forward tilt. Because of this, it can be considered an advantage to have a fast walking speed. This is especially true for characters like Squirtle, Marth and Donkey Kong.
Super Smash Bros. walking speeds
Super Smash Bros. Melee walking speeds
Super Smash Bros. Brawl walking speeds
Super Smash Bros. 4 walking speeds
Super Smash Bros. Ultimate walking speeds
Super Smash Bros. for Wii U digital manual description