Hyperfocal distance is a specific application of the concept of depth of field.
There's only one distance that is in sharpest focus. Everything in front of or behind that distance is blurry. The further we move away from the focus distance, the blurrier things get. The questions become: "How blurry is it? Is that within our acceptable limit? How far from the focus distance do things become unacceptably blurry?"
What we call depth of field (DoF) is the range of distances in front of and behind the point of focus that are acceptably blurry so that things still look like they are in focus.
The amount of depth of field depends on two things: total magnification and aperture. Total magnification includes the following factors: focal length, subject/focus distance, enlargement ratio (which is determined by both sensor size and display size), and viewing distance. The visual acuity of the viewer also contributes to what is acceptably sharp enough to appear in focus instead of blurry.
The distribution of the depth of field in front of and behind the focus distance depends on several factors, primarily focal length and focus distance.
The ratio of any given lens changes as the focus distance is changed. Most lenses approach 1:1 at the minimum focus distance. As the focus distance is increased the rear depth of field increases faster than the front depth of field. There is one focus distance at which the ratio will be 1:2, or one-third in front and two-thirds behind the point of focus.
At short focus distances the ratio approaches 1:1. A true macro lens that can project a virtual image on the sensor or film that is the same size as the object for which it is projecting the image achieves a 1:1 ratio. Even lenses that can not achieve macro focus will demonstrate a ratio very near to 1:1 at their minimum focus distance.
At longer focus distances the rear of the depth of field reaches all the way to infinity and thus the ratio between front and rear DoF approaches 1:∞. The shortest focus distance at which the rear DoF reaches infinity is called the hyperfocal distance. The near depth of field will very closely approach one half the focus distance. That is, the nearest edge of the DoF will be halfway between the camera and the focus distance.
We must also remember that hyperfocal distance, like the concept of depth of field upon which it is based, is really just an illusion, albeit a rather persistent one. Only a single distance will be at sharpest focus. What we call depth of field are the areas on either side of the sharpest focus that are blurred so insignificantly that we still see them as sharp. Please note that the hyperfocal distance will vary based upon a change to any of the factors that affect DoF: focal length, aperture, magnification/display size, viewing distance, etc.
All those calculators and apps always use the value of 0.03mm for the circle of confusion. Why 0.03mm?
Because they assume a format (sensor or film) size of 36x24mm and an enlargement to 8x10 inches (or 8x12") viewed from a distance of 10-12 inches by a person with 20/20 vision. Some lens makers assumed the viewer has 20/15 vision and thus they use a CoC of 0.025 mm instead of 0.03 mm.
The image will appear perfectly sharp when viewed with an angle of view of 50° but when calculating the hyperfocal distance with a circle of confusion diameter limit of 4.34 µm, we should focus at 18.53m which is more than 6 times the hyperfocal distance.
Assuming we're using a 96 ppi monitor, such as a 24" FHD (1920x1080) one, when we view an image at 100% (1 image pixel = 1 screen pixel) from a FF camera with 4.34µm pixel pitch, we're enlarging that 46 MP image by a factor that would result in a viewing size of 86x57 inches! Even accounting for the fact that our eyes are probably more than 10-12 inches from our monitor, that's still a much larger magnification ratio that viewing an 8x10" from 12". Blur that is too small to tell apart from a point at standard viewing conditions (8x10" viewed from 12" by a person with 20/20 vision) will be easy to see when enlarged to 86x57!
The more you enlarge an image, the more you magnify the blur and things that look sharp at smaller sizes gradually become more blurry as we increase the magnification.
As the enlargement ratio increases, the depth of field decreases, and this we must move the focus point progressively further back to keep infinity with the rear depth of field.
For more, please see:
Why did manufacturers stop including DOF scales on lenses?
Is there a 'rule of thumb' that I can use to estimate depth of field while shooting?
How do you determine the acceptable Circle of Confusion for a particular photo?
Find hyperfocal distance for HD (1920x1080) resolution?
Why I am getting different values for depth of field from calculators vs in-camera DoF preview?
As well as this answer to Simple quick DoF estimate method for prime lens