ideal
An ideal
I is represented by its generators. We use the function
ideal to construct an ideal.
i1 : R = QQ[a..d];

i2 : I = ideal (a^2*bc^2, a*b^2d^3, c^5d)
2 2 2 3 5
o2 = ideal (a b  c , a*b  d , c  d)
o2 : Ideal of R

monomial ideals
For a monomial ideal you can use the function
monomialIdeal.
i3 : J = monomialIdeal (a^2*b, b*c*d, c^5)
2 5
o3 = monomialIdeal (a b, c , b*c*d)
o3 : MonomialIdeal of R

The distinction is small since a monomial ideal can be constructed using
ideal . However, there are a few functions, like
primaryDecomposition that run faster if you define a monomial ideal using
monomialIdeal.
monomialCurveIdeal
An interesting class of ideals can be obtained as the defining ideals in projective space of monomial curves. For example the twisted cubic is the closure of the set of points
(1,t^1,t^2,t^3) in projective space. We use a list of the exponents and
monomialCurveIdeal to get the ideal.
i4 : monomialCurveIdeal(R,{1,2,3})
2 2
o4 = ideal (c  b*d, b*c  a*d, b  a*c)
o4 : Ideal of R
