There are, I think, three main senses.
The first sense (sometimes called "true" infinitesimals) is when you have an easily identified collection of things that are not infinitesimal, and some sense of comparing "size". The infinitesimals are those objects that are smaller than every non-infinitesimal.
A typical example is the hyperreals from nonstandard analysis: an infinitesimal hyperreal is a number whose magnitude is smaller than the magnitude of every nonzero (standard) real number.
The second sense is somewhat metaphorical; where you have objects that represent some infinitesimal-like notion. So while you don't have any "true" infinitesimals, you can still use the metaphor to do many of the things you wanted to use infinitesimals for anyways.
For example, the real line has no (nonzero) infinitesimals, but we can talk about its tangent bundle: the set of pairs of real numbers $(x,y)$ where $x$ denotes a point on the real line and $y$ is imagined as the scale of some infinitesimal displacement from $y$. Then, to do calculus with these, we say that if $f$ is a differentiable function, we also treat it as a function on the tangent bundle too, with $f(x,y) = (f(x), f'(x) y)$.
This sort of thing is very important to differential geometry.
We can actually make the tangent bundle into an algebraic structure called the dual numbers in a similar fashion to how the complex numbers are defined: we interpret a real number $x$ as the point $(x,0)$, let $\epsilon = (0,1)$. Addition is defined in the obvious way, and multiplication by setting $\epsilon^2 = 0$. (rather than $i^2 = -1$ as we do with complex numbers)
Repeating the above, if $f$ is differentiable, we set $f(x + y \epsilon) = f(x) + f'(x) y \epsilon $. Note the appealing similarity to the notion of a "differential approximation".
In this sort of algebraic setup, we say that $\epsilon$ is a "nilpotent infinitesimal" (to distinguish from "true" infinitesimals). Nilpotent is an adjective that means you get zero by raising it to a sufficiently large power.
The third sense is approximate; "infinitesimal" is used as a shorthand for the idea of being "approximately infinitesimal", which means something is small enough for whatever purpose you need.