What is the difference between Euclid's division lemma and Euclid's division algorithm?

My textbook claims that Euclid's division algorithm depends upon Euclid's division lemma but can you elaborate?

Euclid's Division Lemma states: for $a$, $b\in\mathbb{Z}$, $b\ne0$, there exist unique $q$, $r\in\mathbb{Z}$ such that $$a=qb+r, \qquad 0\le r<|b|\tag{$\star$}$$
We call $q$ the quotient, and $r$ the remainder. The lemma itself can be proved by induction on $a$. Euclid's Division Algorithm is an algorithm to find the greatest common divisor ($\gcd$) of two natural numbers facilitated by repeated use of the Division Lemma until in the last use of $(\star)$ we get a zero remainder and the process terminates with the $\gcd$ given by the last non-zero remainder. Each time the Division Lemma is invoked in the Division Algorithm, the new $a$ and $b$ are the preceding steps $b$ and $r$ respectively.

For example to find $\gcd(267,126)$ the Division Algorithm takes four steps to get to a zero-remainder, each step a usage of the Division Lemma:

• Step 1: Find the remainder when $267$ is divided by $126$. By Euclid's Division Lemma with $a_1=267$, $b_1=126$, we have $$267=2\times126+15\tag{1}$$ and so $q_1=2$, $r_1=15$.

• Step 2: Find the remainder when $126$ is divided by $15$. By Euclid's Division Lemma with $a_2=126$, $b_2=15$, we have $$126=8\times15+6\tag{2}$$ and so $q_2=8$, and $r_2=6$.

• Step 3: Find the remainder when $15$ is divided by $6$. By Euclid's Division Lemma with $a_3=15$, $b_3=6$, we have $$15=2\times6+3\tag{3}$$ and so $q_3=2$, $r_3=3$.

• Step 4: Find the remainder when $6$ is divided by $3$. By Euclid's Division Lemma with $a_4=6$, $b_4=3$, we have $$6=2\times3+0\tag{4}$$ and so $q_4=2$, $r_4=0$. Here the Division Algorithm terminates since $r_4=0$, and $\gcd(267,126)=r_3=3$ is the last non-zero remainder from Step 3.

Putting it simply,

Euclid's Division Lemma is the statement that any non negative integer $n$ can be expressed in $n=aq+b$ form where $0\leq b<q$.

Division algorithm is a method to compute the Greatest Common Divisor of two numbers which is based on repeated application of Euclid's Divsion Lemma until the remainder comes out to be $0$.