In Macaulay2, each free module $F = R^s$ over a ring $R$ has a basis of unit column vectors $F_0, F_1, ..., F_(s1)$. The monomials of $F$ are the elements $m F_i$, where $m$ is a monomial of the ring $R$. In Macaulay2, orders on the monomials of $F$ are used for computing Gröbner bases and syzygies, and also to determine the initial, or lead term of elements of $F$.
The ring $R$ comes equipped with a total order on the monomials of $R$. A total order on the monomials of $F$ is called
compatible (with the order on $R$), if $m F_i > n F_i$ (in $F$) whenever $m > n$ (in $R$). There are many types of compatible orders, but several stand out: term over position up (the default in Macaulay2), term over position down, position up over term, position down over term, and Schreyer orders.
term over position up: $m F_i > n F_j$ iff $m>n$ or $m==n$ and $i>j$
term over position down: $m F_i > n F_j$ iff $m>n$ or $m==n$ and $i<j$
position up over term: $m F_i > n F_j$ iff $i>j$ or $i==j$ and $m>n$
position down over term: $m F_i > n F_j$ iff $i<j$ or $i==j$ and $m>n$
Induced monomial orders are another class of important orders on
F, see
Schreyer orders for their definition and use in Macaulay2.
In Macaulay2, free modules come equipped with a compatible order. The default order is: term over position up. This is called Position=>Up. In the following example, the lead term is $a F_1$, since $a > b$.
i1 : R = ZZ[a..d];

i2 : F = R^3
3
o2 = R
o2 : Rmodule, free

i3 : f = b*F_0 + a*F_1
o3 =  b 
 a 
 0 
3
o3 : R

i4 : leadTerm f
o4 =  0 
 a 
 0 
3
o4 : R

This is the same as giving the monomial order as:
i5 : R = ZZ[a..d, MonomialOrder => {GRevLex => 4, Position => Up}];

i6 : F = R^3
3
o6 = R
o6 : Rmodule, free

i7 : leadTerm(a*F_0 + a*F_1)
o7 =  0 
 a 
 0 
3
o7 : R

Giving Position=>Down instead switches the test above to i < j. In this case the monomial order on F is: m*F_i > n*F_j if m>n or m==n and i<j.
i8 : R = ZZ[a..d, MonomialOrder => {GRevLex => 4, Position => Down}];

i9 : F = R^3
3
o9 = R
o9 : Rmodule, free

i10 : leadTerm(a*F_0 + a*F_1)
o10 =  a 
 0 
 0 
3
o10 : R

If one gives Position=>Up or Position=>Down earlier, then the position will be taken into account earlier. For example
i11 : R = ZZ[a..d, MonomialOrder => {GRevLex => 2, Position => Down, GRevLex => 2}];

i12 : F = R^3
3
o12 = R
o12 : Rmodule, free

i13 : leadTerm(a*F_0 + a*F_1)
o13 =  a 
 0 
 0 
3
o13 : R

i14 : leadTerm(b*F_0 + c^4*F_1)
o14 =  b 
 0 
 0 
3
o14 : R

i15 : leadTerm(c*F_0 + d^2*F_1)
o15 =  c 
 0 
 0 
3
o15 : R

If one wants Position over Term (POT), place the Position element first
i16 : R = ZZ[a..d, MonomialOrder => {Position => Down}];

i17 : F = R^3
3
o17 = R
o17 : Rmodule, free

i18 : leadTerm(a*F_0 + a*F_1)
o18 =  a 
 0 
 0 
3
o18 : R

i19 : leadTerm(b*F_0 + c^4*F_1)
o19 =  b 
 0 
 0 
3
o19 : R

i20 : leadTerm(c*F_0 + d^2*F_1)
o20 =  c 
 0 
 0 
3
o20 : R
