How small is an infinitesimal quantity?

When speaking of infinitesimals, I see some mathematicians say that it represents an "extremely small" element, such as an infinitesimal area on a manifold.

What bothers me about this naive definition is how small is, an infinitesimal area, for example, supposed to be for it to be called infinitesimal?

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.

In physics, the notion of an infinitesimal quantity or area is used extremely informally, to indicate roughly anything much smaller than some given reference quantity. To give a mathematically acceptable description of the infinitesimal is a more serious undertaking.

"Infinitesimal" means, formally, "Smaller than any positive ordinary number." Practically, that means smaller than any positive number you can explicitly name, such as $.1,.01,.001,...$. This seems paradoxical, and was taken historically, as in the philosophical work of Berkeley in the 18th century, to indicate that the very notion of infinitesimal is incoherent. While these arguments had a significant influence on the formalization of the subject of analysis in the late 19th century, they were not the last word, as Robinson in the 1960s rehabilitated the notion of infinitesimal back into mathematical respectability. Vaguely, the starting point is to observe that there's nothing immediately contradictory about positing the existence of a positive number smaller than any (say) positive rational number, and to axiomatize your way into a situation where such numbers do indeed exist. The most traditional way to introduce such numbers is the study of hyperreal number systems. But for geometrical applications, the theory of synthetic differential geometry may be more useful.

Infinitesimals were used in the genesis of analysis which was appropriately called at the time infinitesimal analysis or infinitesimal calculus. Infinitesimals were used fruitfully for several centuries. Around 1870 certain foundational developments led to the mathematicians jettisoning the infinitesimals. The language they used still exploited the intuitive terminology of infinitesimals but having no precise mathematical counterpart for them, they had to be eliminated when their arguments were formalized.

This resulted in the notorious epsilon-delta paraphrases, the nightmare of a majority of undergraduate math majors. In 1961 infinitesimals were restored to respectability by Abraham Robinson but old habits die hard. The precise meaning of an infinitesimal $\epsilon$ is a fixed number that is less than $\frac12$, less than $\frac13$, less than $\frac14$, all the way down.

Since you mentioned "manifolds" I would like to point out the existence of an approach to differential geometry via infinitesimals where one can naturally talk about infinitesimal area elements (rather than paraphrases in terms of differential forms); see this recent publication for details.

Engineers must use a practical concept of "infinitesimal" because we work with both very large and very small numbers. To us, "infinitesimal" is basically the same as "irrelevant," "ignorable," or "existing within the limit of statistical noise," which we often define generally as less than 3% of the whole. Thus, (a) an infinitesimal fraction of 2pF (two pico-Farads) is anything less than (0.03)*2pF or 60fF (60 femto-Farads) and (b) an infinitesimal fraction of 2MV (two mega-volts) is (0.03)*2MV or 60KV (60 kilo-volts).