You are looking for Lexicographical Order and Monomial Order.
Lexicographic ordering and Degree Ordering
There are 2 phrases involved in (1)-(4): Lexicographic and Degree.
Firstly, each of these terms $4xy^2z,4yz^2,-5x^2,7x^2y^2$ is called a monomial.
Each monomial has a degree, which is just sum of the degrees of individual variables.
deg$(4xy^2z)=4,$ from $x=1,y^2=2,z=1$
deg$(4yz^2)=3$
deg$(-5x^2)=2$
deg$(7x^2y^2)=4$
For Lexicographical ordering, you need to list a sequence first.
For example, we can have the usual alphabetical order: $a,b,c,\dots,x,y,z$
Then this sequence gives you a rule to compare the variables:
i.e. $a>b>c>\dots>x>y>z$
For example, $-5x^2>4yz^2$ because $x>y$ and $x>z$.
Note that the ordering is by degree of $x$ followed by $y$, then $z$.
Example: $7x^2y^2>-5x^2$, since they break even in degree of $x$, but ordering by degree of $y$ distinguishes them.
On the other hand, ordering by degree basically just means that higher degree monomials are greater.
For example, $4xy^2z>4yz^2>-5x^2$ since degrees: $4>3>2$.
Explanation of (1)-(4)
(1) Lexicographical:
Order the monomials by the original listed Lexicographical order.
Order by degree of $a$, then by $b$, etc...
(2) DegreeLexicographical:
Order by total degree first.
For monomials with same degree, distinguish by Lexicographical order (1).
(3) ReverseLexicographical:
Compare backwards: Compare degree of $z$, then by degree of $y$, etc...
If degree is bigger, monomial is smaller
(4) DegreeReverseLexicographical:
Order by total degree first.
For similar degree monomials, compare backwards: compare degree $z$, then $y,\dots$
If degree is bigger, monomial is smaller.
Mathematica Example for (1)-(4)
$f=4 x y^2z+4 y z^2-5x^2+7x^2y^2;$
Case (1):
Command: MonomialList[$f,\lbrace x,y,z\rbrace$,"Lexicographic"]
Result: $\lbrace 7 x^2 y^2,-5 x^2,4 x y^2 z,4 y z^2\rbrace$
Explanation:
$7x^2y^2,-5x^2>4xy^2>4yz^2$ by degree of $x$.
Then $7x^2y^2>-5x^2$ by degree of $y$.
Case (2):
Command: MonomialList[$f,\lbrace x,y,z\rbrace$,"DegreeLexicographic"]
Result: $\lbrace 7 x^2 y^2,4 x y^2 z,4 y z^2,-5 x^2\rbrace$
Explanation:
$7x^2y^2,4xy^2z>4yz^2>-5x^2$ by total degree ordering.
$7x^2y^2>4xy^2z$ by Lexicographic ordering: i.e. degree of $x$.
Case (3):
Command: $f$
Result: $-5 x^2+7 x^2 y^2+4 x y^2 z+4 y z^2$
Explanation:
Mathematica uses Reverse Lexicographic order by default, hence the command.
$4yz^2<4xy^2z<-5x^2,7x^2y^2$ by degree on $z$ (comparing backwards).
Notice that bigger degree in $z$ means smaller monomial.
Then $7x^2y^2< -5x^2$ by degree on $y$ (same degree of $z$).
Bigger degree of $y$ means smaller monomial.
Case (4):
Command: MonomialList[$f,\lbrace x,y,z\rbrace$,"DegreeReverseLexicographic"]
Result: $\lbrace 7 x^2 y^2,4 x y^2 z,4 y z^2,-5 x^2\rbrace$
Explanation
Order by total degree first: $7x^2y^2,4xy^2z>4yz^2>-5x^2$ since $4,4>3>2$.
Then compare degree of $z$ due to reverse Lexicographical ordering.
$4xy^2z<7x^2y^2$ since bigger degree of $z$ means smaller monomial.
You can find a similar rule at Singular and a few more examples at Sage.