I got a question about infinitesimals in non-standard analysis. If I understand correctly, they are defined to be the number that is closest to zero.
However, at the same time, they satisfy all the properties of real numbers - so for example, let's call such an infinitesimal $\epsilon$. Then $2 \epsilon$ and $3 \epsilon$ are greater than $\epsilon$. And most importantly, if they satisfy the rules known from real numbers, then they should also satisfy the archimedean property that states that it's always possible to give a number closer to zero than given number.
I've heard non-standard analysis simplifies some proofs. Erm, how can we prove theorems about real numbers in a system that includes objects that don't satisfy the properties of real numbers (infinitesimals in this case, because they are defined to be the number nearest to zero, which in fact doesn't exist)?
An infinitesimal is not "the number that is closest to zero". It is "a number which is closer to zero than any standard real number". You are right that in NSA, every number has a whole "cloud" of other numbers infinitely close to it, not just one.
The preceding remark doesn't depend on the NSA framework we use. In the rest of this answer, everything is about the Robinson NSA framework exclusively. This works in a system of numbers called the hyperreals, which includes the real numbers as "standard" numbers as well as additional "nonstandard" numbers.
The hyperreals and reals share properties through the transfer principle. But you must be careful in using it. For example, the hyperreals are not Archimedean in the sense that any positive hyperreal is larger than $1/n$ for some standard natural number $n$. Yet they are Archimedean in the sense that any positive hyperreal is larger than $1/n$ for some hypernatural number $n$ (which could be infinite). The transfer principle says every property of the reals is transferred to the hyperreals in this way, but that you must replace all of your original standard objects (like the natural numbers) with their nonstandard analogues (like the hypernatural numbers). (You also have to work with properties of actual numbers rather than properties of sets of numbers.)
Going in the opposite direction, with the right reformulations of classical definitions, the transfer principle lets you prove a theorem in the hyperreals and get the corresponding statement in the reals for free. In a lot of cases these hyperreal definitions are quite intuitive. For instance a function $f$ is continuous if $f(x+h)-f(x)$ is infinitesimal whenever $h$ is infinitesimal and $x$ is a standard real.
I could make some vague comments about model theory, but I think that is best left to someone with more expertise.