Assuming you mean your question in the practical sense rather than about doing logical foundations... picture in your mind the real numbers: that picture is exactly how the hyperreal numbers look.
I'm not even exaggerating. The hyperreal numbers, along with the rest of the nonstandard model of mathematics they're contained in, is carefully designed to have exactly the same properties that the real numbers do within the standard model.
In fact, in some philosophical approaches to the subject (e.g. how one might interpret internal set theory), it's the hyperreals within the nonstandard model that is actually what mathematicians have been studying for the past few millenia.
Except for a few esoteric applications, one only considers the hyperreals in the context of nonstandard analysis, which is all about making comparisons between standard model and a nonstandard model. One can't get the flavor of NSA by asking "what do the hyperreals look like?" — one has to ask the question "how do the reals and hyperreals compare?".
In the usual formulations, the main distinctive feature is that every real number is also a hyperreal number, and that all of the finite hyperreals can be partitioned according to their standard part.
That is, if $x$ is a finite hyperreal, there is some real $r$ such that $r-x$ is an infinitesimal hyperreal.
Phrased conversely, to each real number $r$ there is a halo of hyperreals surrounding $r$ that are an infinitesimal distance from $r$, and these halos partition the finite hyperreals.
If you are willing to continue on to the extended real numbers (i.e. $\pm \infty$), then the nonfinite hyperreals lie in the halos* of $\pm \infty$ depending on whether they are positive or negative.
Keisler's book uses the analogy of a microscope. At one level, you're studying the standard reals, and at any time you may pick a real and "zoom in" to look at its halo of nonstandard reals with that standard part.
*: Be careful that some sources define halo in a different way so that this statement is no longer true.