Puts emphasis on specifically this thing. Same as in english when you make a word long and higher. Note that it replaces が or を, just like は or も does.
今度こそ勝つぞ！ Next time, I'll win for sure!
この曲こそおれがずっと捜していた曲だ！やっとタイツルが分かった。 this song is the song I have been searching for! Finally I figured out the title.
Very often used with pronouns too
いえいえいえ、私こそありがとうございます。 nononono, I am the one who should be thankful.
からこそ
puts emphasis on a result; 'specifically because S1, S2'.
ここは奇麗な川だからこそ、こんあ美味しいさかなが釣れるんですよ。 specifically because the river is so clean here, you are able to catch delicious fish here.
先輩に色々と教えてもらってからこそ、私はここまで来れたのです。 specifically because of all the things that I had received being taught on me by senpai, I was able to come this far.
君だからこそ出来たんだよ。おれはまでそんな凄いことをできない。 Specifically because it is you, you were able to do it. I can't do anything that amazing yet.