Shiny Pokemon can be obtained by breeding, but in most instances (with one exception), the shininess of the parent doesn't affect the offspring's probability of shininess. The odds of hatching a shiny Pokemon from an Egg vary between scenarios:
In games prior to Generation VI (X/Y and OR/AS), the odds of hatching a shiny without the Shiny Charm are 1/8192. With the Shiny Charm (only applicable to Black and White 2), these odds can be reduced to approximately 1/2731.
From Generation VI onwards, the odds of hatching a shiny without the Shiny Charm are 1/4096. With the Shiny Charm (applicable to all games), these odds can be reduced to approximately 1/1635.
The only instance in which the shininess of a parent can affect the shininess of the offspring is in Generation II. Due to shininess being determined by IVs in that generation and IVs being passed down by breeding, it's possible for the offspring of a shiny Pokemon to have 1/64 odds of being shiny (but only if the offspring's gender is the opposite of the parent's), the highest non-guaranteed odds in any Pokemon game to date.
In short, the shininess of a parent only increases shiny odds for the offspring in Generation II, when shiny mechanics work differently. Parents can pass other traits down to their offspring, however, such as Natures, moves, Poke Balls, and Abilities. For a more in-depth guide, the Bulbapedia article on breeding is a great place to start.