1. Using a starter ensures that you "start", or pitch the correct amount of yeast. There are a few calculators you can use to make sure you do in fact use enough yeast for your brew. Many many people under pitch when they do start fermentation.
2. Dry yeast is just that, dried yeast, which you want to rehydrate with warm water about 1-3 hours before pitching into your wort. Liquid yeast is in a liquid and does not require rehydrating. But it is a good idea to use a starter as mentioned above.
3. Washing yeast is when a beer is done and there is all the yeast and trub at the bottom of the fermenter and you "wash" the yeast to seperate the trub from the yeast, which can be reused.