rica ederim is the equivalent of "you are welcome" can be used formally or otherwise bir şey değil = "it's nothing" (direct translation) / "forget about it" / "don't worry about"

You can use both of them... As Amanda says "rica ederim" is a little formal... But "bir şey değil" can be used... I mean it is not absurd for formal...

"Rica ederim" can be used in a wide variety of situations: A: Could you please bring me water? B: Ofcourse. Take it A: Thank you very much. B: You're welcome In this paragrag both "rica ederim" and "birşey değil" can be used interchangeably whilst "rica ederim" is more polite and elegant. But we can olsa use "hiç önemli değil", "aman canım ne önemi var", "bahsetmeye değmez", "ben teşekkür ederim" the last one "(ben teşekkür ederim") is a bit strange; we found an intelligent excuse as if we would already do the job anyway; like in the example above "Asıl ben teşekkür ederim, ne zamandır oturuyordum biraz hareket etmeye ihtiyacım vardı". But we can use "rica ederim" in other situations: Ahmet rica ederim artık şu sigarayı bırak. Ahmet please give up smoking from now on. Ahmet you made an excellent presentation Rica ederim abartıyorsunuz. (Thank you very much but you are exagarating) Rica edersem bu sandalyeyi alabilir miyim? (Could I please take this chair?) Bence senden harika mafya babası olur. Rica ederim nerden çıkartıyorsun bu saçmalıkları. (you are not agree with the first sentence but denying it in a polite way) So rica ederim means what ever you are going to say you want to be polite and like the last one it can be used for refusing something politely.

'bir şey değil' günlük hayatta daha yaygın kullanılır.

In fact "rica ederim" means "no need to even mention it" but most of people translate it "you are welcome" though it's incorrect.