I have trouble translating Turing machine language, can you help me break down language notation to English? My problem is I don't have many issues with creating a Turing machine state table when given a string such as 01101, my issue arises when I am presented with a problem which requires the Turing machine to recognize the language {0n1n | n ≥ 0}.

I have found this guide on a similar problem however I cannot grasp what is going on and what is required from the language statement. Any help in regards to how to read the specified language effectively would be appreciated.
When $s$ is a symbol and $n$ is a natural number, "$s^n$" means "$s$ repeated $n$&nbsp;times".  So you're being asked to recognize the language of all strings that consist of $n$&nbsp;zeros followed by $n$&nbsp;ones, for any possible value of&nbsp;$n\geq 0$.  That is, $\{\epsilon, 01, 0011, 000111, 00001111, \dots\}$, where $\epsilon$ denotes the empty string (some people write $\lambda$ for the empty string, instead).