是非 [ぜひ] – certainly; definitely; without fail

  • 是 [ゼ・これ・この] – just so; this; right
  • 非 [ヒ・あらず] – un-; non; mistake; negative;

You might also see 是非とも, which still means “by all means” or any number of things along those lines.

I’ve always found the construction of this word a bit odd.  Literally, the characters mean “right-wrong” or something along those lines.  So, I took it upon myself to dive into the etymology of the word a bit, courtesy of http://gogen-allguide.com/se/zehitomo.html

[是] means “correct thing,” while [非] means “unfair; unjust; dishonest.”  Essentially, [是非] is a noun that expresses both good and bad.  Perhaps this is a Japanese version of “for better or for worse… in sickness and in health…”

是非 is a very useful word and you’ll likely hear it a lot.  是非お見逃しなく [ぜひおみのがしなく] – “don’t miss (seeing) it.”  You might hear something like this used in a TV show advertisement, similar to NBC’s slogan “Must-See TV” in America.

Synonyms for 是非, depending on context, could be 必ず [かならず] – certainly; without fail, and どうしても – no matter what happens.