Orbital notation is simply a different means that chemists use to describe the wavefunction for a hydrogen atom. Rather than using the term wavefunction, instead we use the word "orbital".
The orbital notation uses only the n and \( \ell\) quantum numbers.
In this notation we simply state the principle quantum number \( n\) as a number. A letter is used to denote the \( \ell\) term as letters s,p,d...
So let's imagine a hydrogen atom in its lowest energy state. This is the ground state. The electron wavefunction is then described by n=1, \( \ell\) = 0, and \( m_{\ell}\)=0. We could also simply call this "orbital" a 1s orbital. (note: when doing this we ignore any mention of \( m_{\ell}\). We might use light to excite the hydrogen atom to a higher energy state. Let's say we went from 1s to 2p. The excited state could then have quantum numbers n=2, \( \ell\) =1. In the absence of any external perturbations such as a magnetic field all the \( m_{\ell}\) levels are the same. So any \( m_{\ell}\) solution could describe this state.