Either usually goes with an either/or proposition. Examples: "We will either leave on Saturday or on Sunday." "You may have either the chocolate cake or the vanilla ice cream." "We can leave for vacation or we can stay here-- it doesn't matter to me either way." Each is a bit different. Examples: "Each of the children will receive a candy bar." "Each student has a box of colored pencils." Each is generally used to indicate every. Every child will receive a candy bar. Every student has a box of colored pencils.

Each is used to point out an specific thing. Either is when you have to make a choice between a thing an another. I think that's the best way to tell them apart but it's just my opinion :) (For instance, the last example you wrote doesn't sound natural because even if they're only two words you have to say 'each'. In case you want to compare, the sentence would be something like: "You have to make a sentence choosing either this word or that one".)

Each and either have very different meanings. You use either in the meaning of 'any one of the two' so 'with either word' = 'with any (one) of the two words'. Each = Every , so 'with each word' = 'with every word' (there is a small differance, but it's very small).