What are starters and yeast washing?
 

I keep seeing these terms come up but what exactly are they?

Yeast Starters - Home Brewing Wiki

Washing yeast - Home Brewing Wiki

regarding the yeast washing wiki - what does it mean by "Try not to remove any of the part that separated"? Is this the liquid portion that sits on top of the settled yeast?

No. At this point the yeast is still in suspension and the part that is settling out is trub. So you are pouring off the suspended yeast and leaving the trub behind, thus getting only yeast into the new jar.

Aren't most packs of yeast that are available from most local brewstores ready to use? So far we have used dry packets and just used a wyeast smack pack. What type of yeast would need a starter?

Mr Malty Pitching Rate Calculator (make sure you pick "dry" or "liquid") will tell

Basically, every beer needs a starter with liquid yeast. Anything with an OG over about 1.060-1.070 needs it with dry yeast.

If you don't make one under those conditions, you'll be underpitching the needed cell count. That will probably still have a pretty decent end result, but stressed yeast and yeast that's undergoing a lot of reproduction probably won't give you the best possible flavor profile and _might_ result in lower attenuation (hence sweeter, lower ABV beer).

So making a starter to get the right pitching count is important to making a really good beer.

Certain styles of beer rely on either underpitching or overpitching those levels intentionally, but unless you know your beer it's a good guide.

Why go with a liquid yeast if you can achieve the same product with a dry yeast that you can just pitch?

It is not quite the same end result. The different liquid yeasts available can contribute a lot to the final flavor of a brew, whereas most dry yeasts tend to be fairly neutral, and they give less flavor. You can still make mighty fine beer with dry yeast.

If I'm making a beer that needs a pretty neutral yeast or one that's otherwise available dry, I'll go with a dry yeast. There's a reason US-05 is insanely popular.

A lot of what I make (see signature for current examples) relies on more obscure yeasts for the flavor, so I wind up using liquid yeast and making a starter for those.