Find points within radius in a flat plane

Question

I have a 2D flat plane with points. Each point is stored in a database and has it's own x,y cordinates. Example: 2D plane is a 800px * 800px square.

   $points[0]['x'] = 100;
   $points[0]['y'] = 100;
   $points[1]['x'] = 120;
   $points[1]['y'] = 230;
   $points[2]['x'] = 240;
   $points[2]['y'] = 680;
   //More points here
   $points[52]['x'] = 700;
   $points[52]['y'] = 140;

I already have generated the distance of all points using Euclidean distance formula, but as points increases, so the number of calculations that must be done.

So I want at first to limit the number of calculations by searching "neighbor" points only. I can't find or to be more precise, I don't know how to search correctly my question.

What is the algorithm that I must use in order to search within a radius?

A picture with my problem https://i.stack.imgur.com/LONUi.png

Black point is the point from which I want to find the distance on red points within the radius. I am using PHP for this specific problem.

P.S. No coordinates, latitude/longitude.