Oh, okay. I just read through your entire question. What easybib spits out is going to be what you’ll use in your Works Cited (aka your bibliography). What you’re going to write in your actual paper is the following:
Blah blah, this is my paper. Now I shall cite a source, “War does not determine who is right – only who is left” (Russell 23). If I desire to cite this source again, but for example a quote from the next page, I shall write, “I am Bertrand Russell, bad-ass mo-fo. Do not F* with me!” (24). Now view my spectacular analysis of these wondrous quotes.
To sum up:
If you quote the same work two or more times in a row, you don’t put their name again after the first time.
If there is punctuation at the end of your quote: periods go outside the quotation marks; exclamation and question marks, inside.
MrsWho’s answer just popped up, and made me realize I need the disclaimer that I was working under the assumption that you were citing it directly from the book. If you’re citing a source inside of a different source, things get more tricky, and her links are probably going to be useful to you.
PS I also assumed that your quote was on page 23, a number that I made up. You’re probably going to want to change that to whatever page it’s actually on.