What are hyperreal numbers? (Clarifying an already answered question)

The usual construction of the hyperreal numbers is as sequences of real numbers with respect to an equivalence relation. For example, the real number 7 can be represented as a hyperreal number by the sequence $(7,7,7,7,7,\ldots)$, but it can also be represented by the sequence $(7,3,7,7,7,\ldots)$ (that is, an infinite number of 7s but with one 7 replaced by the number 3). Any real number $x$ can be represented as a hyperreal number by the sequence $(x,x,x,\ldots)$. An example of an infinitesimal is given by the sequence $(1,1/2,1/3,1/4,\ldots)$, which happens to be a sequence of numbers converging to $0$. An example of an infinite number in the hyperreals is given by the sequence $(1,2,3,4,\ldots)$.

The equivalence relation is a bit complicated so I won't tell you how it works but just tell you one exists, and that for any sequence of numbers there are many other sequences that correspond to the same hyperreal number. This is analogous to the construction of the rational numbers as numbers of the form $a/b$ where $a$ and $b\neq 0$ are integers, where for instance $2/6$ and $1/3$ are considered equivalent as rational numbers. The equivalence relation for rational numbers is quite simple, but I'll mention that the equivalence relation for hyperreal numbers is not constructive (it uses the axiom of choice), so it is not in general possible to tell whether two sequences are equivalent as hyperreal numbers.

When actually working with hyperreal numbers however, how they were constructed is not important (whether by the method of identifying different sequences of real numbers as above, or otherwise), and the real number 7 is simply called 7 in the hyperreal numbers, and whenever an infinitesimal is needed, one might simply call it $\varepsilon$ with no regard for which specific infinitesimal it is.

Unfortunately, there is no "concrete" description of the hyperreals. For instance, there is no way to give a concrete description of any specific infinitesimal: the infinitesimals tend to be "indistinguishable" from each other. (It takes a bit of work to make this claim precise, but in general, distinct infinitesimals may share all the same definable properties. Contrast that with real numbers, which we can always "tell apart" by finding some rational - which is easy to describe! - in between them; actually, that just amounts to looking at their decimal expansions, and noticing a place where they differ!)

Similarly, the whole object "the field of hyperreals" is a pretty mysterious object: it's not unique in any good sense (so speaking of "the hyperreals" is really not correct), and it takes some serious mathematics to show that it even exists, much more than is required for constructing the reals.

While the hyperreals yield much more intuitive proofs of many theorems of analysis, as a structure they are much less intuitive in my opinion, largely for the reasons above.

To answer your other question, yes, the reals are (isomorphic to) a subset of (any version of) the hyperreals; that's what's meant by saying that the hyperreals extend the reals.

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.}