system
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human
ELI5: How do white blood cells "chase" something? Like I get how they move, but how do they know the thing is there without eyes? And how do they make decisions to follow it without a brain?

I've seen videos of white blood cells chasing down various bacterium or whatever and they appear to be distinct organisms with decision-making abilities and whatnot, but surely it's just chemical reactions (and far simpler chemical reactions than our own decision making processes).  

I'm not so much asking how they move, but how they "know" to move, and where to move, and how they "know" what is an invader and what is not.
gpt
They can track them by touch and by smell.

Basically how it works is each of your B lymphocytes (the white blood cells that chase down bacteria) gets randomly assigned one specific antigen (a cell part visible from the outside of the cell) to look out for for and then gets released into your blood to float aimlessly around gloming onto cells and seeing if they match that antigen. (This is also how they know which things are invaders and which aren't: the cells are 'trained' in your bone marrow by basically exposing them to your own cells and killing any of the ones that exhibit an immune response - these ones are defective and would attack your own body, so they get weeded out before they're released into the bloodstream.)

If they ever find a match, that B lymphocyte starts releasing a variety of chemicals into your blood: antibodies (which latch onto the antigens it found and makes the bacteria sticky and visible), and cytokine (which are signalling chemicals which your other white blood cells can 'smell' and follow to the site of the infection, based on the concentration gradient of that particular cytokine). It also starts rapidly dividing and reproducing into a host of B lymphocytes that all match that one antigen, many of which undergo a different developmental pathway (which is based on the presence of those cytokines) to specialize them for different roles in the immunological response.