What's an example of an infinitesimal?
Solution 1:
One example of infinitesimals that has been used historically is that of hornangles, which were particularly popular in the 17th century. A hornangle $\alpha$ can be thought of as the "crevice" at the origin between the $x$-axis and the graph of the parabola $y=x^2$. If a real number $r>0$ is represented geometrically by the angle (in the first quadrant) between the $x$-axis and the line $y=r x$, it is easy to convince oneself that $\alpha$ is less than $r$ because a sufficiently short arc of the parabola $y=x^2$ will necessarily dip below the line $y=rx$ (when $r$ is fixed).
In a more arithmetic vein, Skolem in 1933 used sequences of integers to construct an extended number system incorporating infinite numbers. Here an infinite number is represented by a sequence tending to infinity. Skolem's construction does not rely on the axiom of choice. Using the quotient field of Skolem's integers, one gets a number system where a large fragment of calculus can be treated.
Similarly, Keisler in his book https://www.math.wisc.edu/~keisler/calc.html on page 913 gives an example of an infinitesimal represented by a sequence tending to zero, such as $(\frac{1}{n})$. Here the infinitesimal represented by $(\frac{1}{n^2})$ will be smaller than the infinitesimal represented by $(\frac{1}{n})$, etc.
Classroom experience shows that students find such examples intuitively appealing.
An additional approach is the one using Levi-Civita fields with the lexicographic ordering. These were used by Shamseddine and colleagues to develop computer implementations with infinity; see http://www.bt.pa.msu.edu/index_cosy.htm Needless to say, these "nonstandard models" are completely explicit.
As editor @nombre pointed out in a comment, the transfer principle is the crux of the matter. Depending on the theory one wishes to transfer, the level of difficulty may vary considerably. For example, if the theory is PA then Skolem's construction in ZF (without relying on the axiom of choice) is enough. If one wishes more powerful tools one will need more foundational input. This is a point often overlooked, even by high-profile people like Connes. The issue of constructiveness of Skolem's procedure is probed in more detail in this question: https://mathoverflow.net/questions/227945
Skolem's numbers are relevant because they actually embed in the hyperreals as explained in this article: http://dx.doi.org/10.1007/s10699-012-9316-5
Teaching experience shows that freshmen react well to examples of infinitesimals as represented by null sequences (i.e., sequences tending to zero). They also have some exposure to equivalence relations usually, so they find comprehensible a comment to the effect that an infinitesimal is not exactly a null sequence but rather an equivalence class of those. Of course the hyperreals cannot be constructed in a freshman calculus class any more than the reals.
Solution 2:
The discussion is mostly revolving around Robinson's non-standard analysis, even though the OP has not actually specified what kind of infinitesimals he is looking for. There are definitely good sources on how to teach Robinson's non-standard analysis, but I am not familiar with those, so I cannot say much about that.
Since the purpose is to teach people, it is worth looking at nilpotent infinitesimals, as known in Synthetic Differential Geometry. They quickly give the students methods of calculation that are practically the same as the methods employed by physicists. Synthetic Differential Geometry can be presented axiomatically, without ultrafilters or much attention to the logical language (standard vs. non-standard, internal formulas and). As long as we are interested in concrete computations, we will not even notice the main snag, which is lack of excluded middle.
Here are some references which explain infinitesimals in an accessible way which ought to appeal to students:
I highly recommend John Bell's A primer on infinitesimal analysis, or his shorter An Invitation to Infinitesimal Analysis. If you tone down the stuff about intuitionistic logic and just skip to the axioms and computations, you can get a lot of interesting stuff quickly.
I apologize for blowing my own horn, but I once wrote a blog post about intuitionistic mathematics for physics which has a section about infinitesimal analysis. This was targeted at students of physics. There is a similar exposition by me in the book A Computable Universe.
If you teach computer science students you can motivate infinitesimals through the use of dual numbers in automatic differentiation, a very cool technique for writing programs that automagically calculate derivatives. You don't quite get "true infinitesimals" but it is a start and it can be quite appealing to computer-sciency students.
You asked specifically how to present to the students infinitesimals, or perhaps how to show them "a concrete" infinitesimal. This is always a bit difficult to do, both in Robinson's non-standard analysis and in Synthetic Differential Geometry. In Robinson's theory things revolve about non-principal ultrafilters, which are probably not the sort of thing you want to teach beginning analysis students. In Synthetic Differential Geometry we have (a squre-nilpotent infinitesimal is defined to be an element of the smooth real line $R$ whose square is zero).
$0$ is not the only square-nilpotent infinitesimal: $\lnot \forall x \in R . (x^2 = 0 \Rightarrow x = 0)$.
No square-nilpotent is distinct from zero: $\lnot \exists x \in R . x^2 = 0 \land (x < 0 \lor x > 0)$.
These two statements taken together are quite counter-intuitive, especially for a person who is used to classical logic (so 99.99% of mathematicians). The second statement actually tells us that we cannot ever display a concrete infinitesimal that is detectably different from $0$. Infinitesimals are intrinsically strange! But you can actually take advantage of the oddity and entice students to question some basic assumptions about how their geometric intuitions work and what sort of things are possible in mathematics. It is a lot of fun. I tried to explain the strange status of infinitesimals in my blog post and the paper, so I will not repeat that here.
Solution 3:
I am teaching a calculus class using infinitesimals, so this question is very relevant for me too, user72694. I am avoiding formal constructions with ultrafilters and such, because I have found that these are too abstract for my students. What I want is something more concrete that will allow the students to build a reasonably robust concept image for infinitesimals -- something they can appeal to as necessary when they start working with dxs.
So the examples I give them are 0.000...1 (infinity 0s followed by a 1), 0.000...2, 0.000...01, etc. The reason I like these is that in my research students are able to independently reason with them and to develop conjectures about what happens when you add them, multiply, take ratios, square them, etc.
Of course, if someone asks me what these really are, then I will provide a more formal explanation for using sequences (keeping in mind that research shows they don't usually even understand what 0.999... is at this point). Then I'll talk about how you can think of these infinitesimal decimals as specific sequences of numbers that converge to 0 (there are others like 1, 1/2, 1/3, 1/4, ... that I don't know how to write as "decimals"). Ultrafilters etc. arise only when a student figures out that it's actually a bit hairy to compare two convergent sequences where one doesn't strictly dominate the other.