A term used extensively in uses of nonstandard analysis (I shall use the concepts from that set of notes, so it may be useful either to read them or be otherwise familiar with Abraham Robinson's formulation of nonstandard analysis).
I describe first the concept in a metric space (say R, the real numbers, or any vector space R^{n} with the Euclidian norm or another norm, or whatever you like), then in greater generality in a topological space. Since the concept is topological, not metric, metrics which give the same topology on the space will give the same concept of standard approximation.

Suppose X is a (standard) metric space. Then X^{*} is the corresponding nonstandard version of X, and X^{*}\X is the set of nonstandard elements in the content of X.
Suppose furthermore than x is a standard member of X. Form the formulae "d(x,y)<ε" for all standard ε>0 (if it makes you feel better to have just countably many formulae, you could just as well use the formula "d(x,y)<1/n" for all standard natural numbers n). A member (as we'll immediately see, it's always a pseudomember) y≠x of X^{*} which satisfies all these formulae is called "infinitesimally close to" x. In this case, x is known as the standard approximation of y.

Suppose X is a (standard) topological space which is a Hausdorff space. Again, X^{*} is the corresponding nonstandard version of X.
If x is a standard member of X, we form the formulae "y∈U_{x}" for all neighborhoods U_{x} of x. A member y≠x of X^{*} which satisfies all these formulae is called "infinitesimally close to" x. Again, y will of necessity be nonstandard, and x is called the standard approximation of y.
Note that any standard x is its own standard approximation.
There is at most one standard approximation to any y in X^{*}. In the metric case, we note that if x and z are both standard approximations to y, then d(x,y)<1/n and also d(z,y)<1/n for all n. So from the axioms for the metrics we have d(x,z)≤d(x,y)+d(z,y)<2/n. But x,z are standard, and therefore d(x,z) must also be standard; it follows that d(x,z)=0, and x=z.
In the topological (but Hausdorff) case, suppose y∈U_{x} and y∈V_{z} for all neighborhoods U_{x} of x and V_{z} of z. If x≠z, take standard disjoint neighborhoods U of x and V of z (they exist, by the Hausdorff property of X) and apply them. Then y is a pseudoelement of U∩V, but also the formula "U∩V=∅" is true in the standard world, therefore true in the nonstandard world too, and therefore U^{*} and V^{*} can have no common elements. So we must have x=z.
Existence
If X is finite, then it has no nonstandard pseudoelements, and standard approximation is not a useful property.
If X is infinite, then some of its elements may be standard approximations of nonstandard pseudoelements. But a nonstandard element of X^{*} need not, in general, have a standard approximation. Indeed, it may happen that no nonstandard element of X^{*} has a standard approximation. For instance, if X=N (the natural numbers) and y is a nonstandard pseudoelement of X, y has no standard approximation: for any standard element x of N, it is true that "∀y∈N.d(x,y)<1 → x=y". In particular, if x were the standard approximation of y, then certainly "d(x,y)<1" would hold, and therefore also "x=y", meaning y=x is standard.
If X=R (the real numbers), then some nonstandard pseudoelements of R have a standard approximation and some do not. Let r be the set of nonstandard elements whose standard approximation is 0. Then for any standard real x, the set of nonstandard elements whose standard approximation is x is r_{x}=x+r. Proving this is a technical exercise.