A good question, with a fascinating history.
I'll answer from the perspective of electronic
materials in materials science. In short, with
today's experimental techniques, we can see
atoms! In fact, we can even manipulate
individual atoms for potentially useful things
(more on this later)!
But this was not always the case. One of the
earliest records we have on the atom came from
Democritus, an ancient Greek philosopher (others
like Plato and Aristotle had similar trains of
thought). Democritus had a thought experiment. The
idea was if you took a material and divided it
half, you would have a smaller but identical
chunk. If you keep dividing your material, there
should eventually be a point where you've reached
the smallest representative element of your
material. That element is the"atom". In
fact "atom" is derived from the Greek word
"atomos," which roughly translates to indivisible
(it turns out there are even smaller components
that make up an atom, but the name stuck; see
Standard Model ).
It wasn't until much later in the 18th and 19th
centuries that significant progress towards
understanding the atom was made. Our understanding
of the structure of the atom has vastly changed
from the solid-ball model John Dalton first
proposed with the development and advancement of
quantum mechanics. It is the interplay between
theory (e.g., quantum mechanics) and experiment
that let's us characterize and engineer materials
at the atomic level. Both are part of a feedback
loop where e.g., experiment confirms a theoretical
prediction or theory explains what is observed
Read more about the history and how our
understanding of the atom changed
here ) (several seminal experiments that
contributed to this are described).
The coolest part is the fact that we can see
atoms in experiment, and even manipulate them!
Here is the famous image of 35 Xenon
atoms precisely positioned to form the letters
'IBM' using a scanning tunneling microscope.
Each dot you see is an individual Xenon
atom. Read more
Research on engineering materials at the
atomic level is a fast growing area (you might
have heard of nanotechnology, for instance). For
example, here at UCSB, there is active research on
growing materials with atomically sharp
interfaces. Here is an
example . Each of the balls you see
corresponds to a La or Sr atom; the
other elements are harder to see because they are
smaller/lighter, though you might be able to see
the >b>Al and Ti if you squint hard
enough). This material was carefully grown using a
technique called molecular beam epitaxy
(MBE), and is being studied for its rich
physics and as the next big electronic material
for things like your computer and phone.
Hope this helps!