Soft resetting (often shortened as SRing) is a feature found in the main series Pokémon games, with the exception of Pokémon Colosseum. It involves pressing certain buttons on the game system to reload the game from the startup screen.
This practice is widely considered useful, as rebooting the game will
modify hidden data values. The values that determine the shininess of
an in-game event Pokémon is of particular interest here, as are the
Pokémon's IVs and Natures. Soft resetting eliminates the need to
constantly turn off (and on) the game when attempting to get a Pokémon
with these specific qualities. Although time-consuming, it is a widely
practiced method to get a desired Pokémon.
So basically, it's where you save and press a certain combination of buttons (such as in DS games, you'd press and hold L, R, Start, and Select) to restart the game and re-encounter the Pokemon. And you do this to get the Pokemon you have in mind—with the right nature, or a shiny one.