What we mean by that is a list of useful particular cases of expressions like
$$aa^{\dagger k}a^la^\dagger=a^{\dagger(k+1)}a^{l+1}+(k+l+1)a^{\dagger k}a^l+kla^{\dagger(k-1)}a^{l-1}$$
and still other more cases that I encounter frequently in computations, but also for $b$ in which case the indices are $m$, $n$ instead of $k$, $l$. With the following list I can just copy/paste results to speed up tedious quantum algebra.
With three operators:
$$a\ud{a}^k a^l=\ud{a}^ka^{l+1}+k\ud{a}^{k-1}a^l$$
$$\ud{a}^ka^l\ud{a}=\ud{a}^{k+1}a^l+l\ud{a}^ka^{l-1}$$
and particular cases often encountered:
$$a\ud{a}a\ud{a}^k=\ud{a}^{k+1}a^2+(1+2k)\ud{a}^ka+k^2\ud{a}^{k-1}$$
$$a^l\ud{a}a\ud{a}=\ud{a}^2a^{l+1}+(1+2l)\ud{a}a^l+l^2a^{l-1}$$
$$aa^\dagger a^{\dagger k}a^{l}=a^{\dagger(k+1)}a^{l+1}+(1+k)a^{\dagger k}a^l$$
$$a^{\dagger k}a^{l}aa^\dagger=a^{\dagger(k+1)}a^{l+1}+(1+l)a^{\dagger k}a^l$$
$$a^\dagger aa^{\dagger k}a^{l}=a^{\dagger(k+1)}a^{l+1}+ka^{\dagger k}a^l$$
$$a^{\dagger k}a^{l}a^\dagger a=a^{\dagger(k+1)}a^{l+1}+la^{\dagger k}a^l$$
$$(a^\dagger a)^2a^{\dagger k}a^l=a^{\dagger(k+2)}a^{l+2}+(1+2k)a^{\dagger(k+1)}a^{l+1}+k^2a^{\dagger k}a^l$$
$$a^{\dagger k}a^l(a^{\dagger}a)^2=a^{\dagger(k+2)}a^{l+2}+(1+2l)a^{\dagger(k+1)}a^{l+1}+l^2a^{\dagger k}a^l$$
$$bb^{\dagger m}b^nb^\dagger=b^{\dagger(m+1)}b^{n+1}+(m+n+1)b^{\dagger m}b^n+mnb^{\dagger(m-1)}b^{n-1}$$
$$b^\dagger bb^{\dagger m}b^{n}=b^{\dagger(m+1)}b^{n+1}+mb^{\dagger m}b^n$$
$$b^{\dagger m}b^{n}b^\dagger b=b^{\dagger(m+1)}b^{n+1}+nb^{\dagger m}b^n$$
$$b\ud{b}^mb^n=\ud{b}^mb^{n+1}+m\ud{b}^{m-1}b^n$$
$$\ud{b}^mb^n\ud{b}=\ud{b}^{m+1}b^n+n\ud{b}^mb^{n-1}$$