This is a potentially very very broad question, but I'll try to provide a simple answer that addresses the biggest misconceptions.
First of all, animal vision (and brains more generally) is massively parallel. There may be some serial processing steps, but these are also massively parallel. Computers need to digest information into stereotyped operations that can be executed on a CPU. The brain has separate dedicated machinery to each point in space for early vision, so there is no need to process individual "lines" or points in sequence: it all happens at once.
Photoreceptor inputs are converted into center-surround receptive fields in retinal ganglion cells, where light in the center excites and light in the surround suppresses (ON-center cells), or vice versa (OFF-center cells). These receptive fields are then transmitted to thalamus (the lateral geniculate nucleus), and from there to V1, primary visual cortex.
You can then combine many of these circular receptive fields to detect straight edges, like this:
The cells in V1 that respond to these "edges" are called "simple cells"; there are also "complex cells" that have more complicated receptive fields, and other sensitivity like to motion and color. Some computer vision strategies end up producing receptive fields that look a lot like the ones found in early visual areas, the earliest ones built out of Gaussian-filtered sin waves.
From V1, there are higher order visual areas that respond to things like shapes, motion, optic flow, etc.
Basic neuroscience textbooks tend to contain a lot of information on the early visual system, Purves Neuroscience is a good example, any edition is fine:
Purves, D., Augustine, G. J., Fitzpatrick, D., Hall, W. C., LaMantia, A. S., McNamara, J. O., & White, L. E. (2014). Neuroscience, 2008. De Boeck, Sinauer, Sunderland, Mass.