Infinitesimals are merely the (non-standard) objects whose standard approximation is 0.
In other words: Form the set Φ(x) of all formulae
0 < x & x < 1/n
for every (standard!)
natural numbers n.
As R is archimedean, you could equally well take all rational numbers or all real numbers n, it makes no difference.
This set is finitely satisfiable in the real world, hence it is satisfiable in the non-standard world. Any x satisfying all of Φ(x) is a (positive) infinitesimal.
Let O be the non-standard set of all infinitesimals. As (R,+) is a topological group, it turns out that the standard approximations for any real a are precisely a+O -- take a, and add an infinitesimal. This meshes nicely with our intuitions.
The notion of standard approximation is thus easier to use than infinitesimals (it also works in any metric space). Ironically, Robinson's non-standard analysis tends not to use infinitesimals by their name, but rather that wider related concept.