There is a repository for that: https://github.com/vjudeu/curves1000/tree/master/png

Everything up to 1000 is covered, for example for p = 967: https://github.com/vjudeu/curves1000/blob/master/png/967.png

You have (x,y) points, where each coordinate is in range from 1 to p - 1. And you have N points. Which means, that instead of 967 x 967 square with 906 white dots, you have 115792089237316195423570985008687907853269984665640564039457584007908834671663 x 115792089237316195423570985008687907853269984665640564039457584007908834671663 square with 115792089237316195423570985008687907852837564279074904382605163141518161494336 white dots inside.

Technically, you can represent it in many different ways, but in general, you have just (x,y) points in a huge space. Which is why around 2^128 operations are needed to break it.

I think the easiest way to understand it, is to start with smaller examples, and to increase numbers gradually, until you get it. And again, there is even another topic about it: https://bitcointalk.org/index.php?topic=5459153.0