How do we see individual atoms?
Not with light — atoms are far smaller than the wavelength of visible light, so no optical microscope can ever resolve them. The images you have seen come from instruments that feel surfaces or use electrons instead of photons.
The fundamental limit. Optical resolution is bounded by wavelength: visible light spans roughly 400 to 700 nanometres, while an atom is around 0.1 nanometres. Using light to image an atom is like trying to feel the texture of sandpaper with a boxing glove.
Scanning tunnelling microscopy. An atomically sharp conducting tip is brought within a nanometre of a conducting surface. Electrons tunnel quantum-mechanically across the gap, producing a current that varies exponentially with distance — so extraordinarily sensitively that moving a fraction of an atom's width changes it measurably. Scanning the tip and recording the current maps the surface atom by atom. It won a Nobel Prize in 1986, and famously allowed individual atoms to be moved deliberately and arranged into patterns.
Atomic force microscopy works similarly using force rather than current, so it images non-conducting materials and biological samples too.
Electron microscopy. Electrons have wavelengths thousands of times shorter than visible light. Transmission electron microscopes fire them through thin samples, and modern aberration-corrected instruments resolve individual atomic columns and can identify elements from how electrons scatter.
X-ray crystallography, which does not produce a picture at all: it measures how X-rays diffract from a crystal and computes the atomic positions from the diffraction pattern. This is how DNA's structure and most protein structures were determined.
What these images actually are. Not photographs. They are maps of a measured quantity — tunnelling current, force, electron scattering — rendered visually and usually false-coloured. An atom has no colour and no sharp edge; what is drawn is a probability distribution of electrons.
The honest caveat: interpretation depends on the model used, and artefacts from tip shape or sample preparation are a genuine and well-known hazard.