Cellular Automata
It is hard to describe what a cellular automaton really is; I have learned this fact through my many (failed) attempts at explaining what these dots on my screen represent to a friend. Maybe I should start with the most well-known example: Conway's Game of Life (CGoL).
In the Game of Life, cells lie on a square grid, each with a binary state: ON or OFF. At each discrete time step, each cell updates its state according the count of ON states in a neighborhood of cells, and all cells update at the same time according to the previous "snapshot" of the grid.
Run a given initial configuration for enough iterations (or generations) and stuff appears: patterns that stay invariant every generation (still lifes); patterns that repeats after a number of generations (oscillators); patterns that move to a different location after some generations (spaceships); patterns whose number of on cells (population) grows linearly, quadratically or logarithmically; and many more.
Cellular automata (and CGoL itself) have been used to do many things. To list a few: simulating biological systems and physical systems (e.g. gas models), calculating primes (one early design by me and a much better design by someone else), simulating itself (see metacell). No wonder Wikipedia calls it a "model of computation"!
As you can see, people have made great interesting things with cellular automata. But note that that ones I've listed were all intentionally designed, be it patterns or the actually update rules. I personally like waste my free time on what the community calls rule-golfing, i.e. exploring the rule space by random walk. If you relax CGoL's update rule from count of ON states in its neighbor to their exactly arrangement, while maintaining rotational and reflectional symmetry, then you get about 2^102≈5.07e30 different rules. Inside this giant rule space, I usually start with someone else's interesting rule, then make random changes to it until I observe interesting behavior in the evolved configurations. The rules and objects I end up with are really discovered rather than constructed.
Here I list the particularly cool-looking, unusual, or otherwise notable discoveries by me. The gibberish text encoding the rule and the pattern in RLE format, which you can simulate yourself using e.g. Golly.
Jellyfish
x = 17, y = 17, rule = B2n3ai4eirt5cekqr6ce7/S2cn3-ajr4ciknqtz5-anry6-a7c8
2bo$b3o$5o$b5o$2bob3o$5b3o$5b4o$3b7o$4b7o$5b7o$6b7o$7b7o$8b2o2b3o$9bo
3b3o$12b5o$13b3o$14bo!
Spiral
x = 8, y = 13, rule = B3aeiqr4-aijn5c6cei7/S2cn3-ajr4ceiqt5eijkq6-a7c8
2bo$b3o$5o$b5o$2b5o$3b5o$2b5o$b5o$5o$4o$3o$2o$o!
Cross ship
x = 108, y = 75, rule = B3airy4-jkrw5ceikq6-an7/S2ckn3-ajqy4cnqrz5cejkq67c8
7bo59bo$6b3o57b3o$5b5o55b5o$4b7o53b7o$3b9o51b9o$2b6ob4o49b4ob6o$b10o
53b10o$9obo53bob9o$b4ob3o57b3ob4o$2b5o2bo55bo2b5o$3b5o2bo53bo2b5o$4b2o
5bo51bo5b2o$5bo6bo49bo6bo$13bo47bo$14bo45bo$15bo43bo$16bo41bo$17bo39bo
$18bo37bo$19bo35bo$20bo33bo$21bo31bo$22bo29bo$23bobo23bobo$24b2o23b2o$
23b3o23b3o$26bo21bo$27bo19bo$28bo17bo$29bo15bo$30bo13bo$31bo11bo$32bo
9bo58b3o$33bo7bo58bobobo$34bo5bo58bob3obo$35bobobo58bob2ob2obo$36b3o
54bo3bob7obo$35b5o57b3ob3ob3o$36b3o54bo3bob7obo$35bobobo58bob2ob2obo$
34bo5bo58bob3obo$33bo7bo58bobobo$32bo9bo58b3o$31bo11bo$30bo13bo$29bo
15bo$28bo17bo$27bo19bo$26bo21bo$23b3o23b3o$24b2o23b2o$23bobo23bobo$22b
o29bo$21bo31bo$20bo33bo$19bo35bo$18bo37bo$17bo39bo$16bo41bo$15bo43bo$
14bo45bo$13bo47bo$5bo6bo49bo6bo$4b2o5bo51bo5b2o$3b5o2bo53bo2b5o$2b5o2b
o55bo2b5o$b4ob3o57b3ob4o$9obo53bob9o$b10o53b10o$2b6ob4o49b4ob6o$3b9o
51b9o$4b7o53b7o$5b5o55b5o$6b3o57b3o$7bo59bo!
Alien Glyph
In rule B2i3aei4aceny5aceny6i/S2ei3ijk4acijqw5-cn6ae7e8
Moving Spirals
x = 42, y = 45, rule = B2i3aeikq4acqtyz5cery6i7e/S2ekn3ij4acinqrw5-c6ae7e8
30bo$28b2o$26b3o$24b4ob2o$22b3obo$20b4ob2o2bo$19b2obo2bo$18b2ob2o2bo$
18bobo$17b4o$17b2o$16b2o14bo$3b4o9bobobo9b5o$3bo4bobo4b3o11bo3b4o$15b
2o13bo3bob2o$bo12b2o8bo9b2ob2o$ob3o9bobo19bobo$2o11b3o20b4o$b2o11bobo
12bo8b2o$b4o9b2o8bo5bo8b2o$2bob3o8b2o12bo7b2obo$2b2o11b3o18b6o$3b2o11b
obobo7bo8b2obo$3b4o9b2o8b5o8b2o$4bob3o8b2o7bobobo7b2o$4b2o11b4o5b2ob2o
5b4o$5b2o11bobo7bo7bobo$5b4o9b2ob2o2bo2bo2bo2b2ob2o$6bob3o8b2obo3bo3bo
3bob2o$6b2o12b4ob2o3b2ob4o$7b2o13b3obo3bob3o$7b4o13b4ob4o$8bob3o13b5o$
8b2o18bo$9b2o$9b3o3bo$10bobo2bo$10b4ob2o2bo$12b3obo2bo$14b4ob2o2bo$16b
3obo$18b4ob2o$20b3ob2o$22b4o$24bo!
Spiral 2
x = 8, y = 12, rule = B3aiq4acjy5ry6i7e/S2ek3cijk4acinqwz5-cn6ae7e8
ob2obo$3ob3o$bo3b2o$2bobob2o$5bobo$6b2o$5b2o$3b4o$2b2obo$b2ob2o$bob2o$
b3o!
Spiral 3
x = 22, y = 21, rule = B3aiq4acy5kry6i7e/S2ek3ijk4acinqw5-cn6ae7e8
bo$2o$o7$13b3o$13bobo$13b4o$15bo6$21bo$20b2o$20bo!
Spiral 4
x = 8, y = 13, rule = B3aeiq4ceikrt5ackq6cen7c/S2-ai3-jqr4ceik5eijkq6-a7c8
2bo$b3o$5o$b5o$2b5o$3b5o$2b5o$b5o$5o$4o$3o$2o$o!
Lines Replicator
x = 27, y = 27, rule = B2i3aei4acey5aceny6i/S2ei3ijk4acijqw5-cn6ae7e8
10b2o$10b4o$7bo3bob3o$8bo3bob4o$14b2ob3o$16bob4o$18b2ob2o$2bo17bob2o$
3bo17bobo$22b3o$2o20b3o$3o21b2o$bobo19bobo$b2o21b3o$2b3o20b2o$2b3o19b
3o$3bobo17bobo$3b2o19b2o$4b3o15b3o$4b3o15b3o$5bobo13bobo$5b2obo11bob2o
$6b2ob2o7b2ob2o$7b4obo3bob4o$9b3obobob3o$11b7o$13b3o!
Oscillator
x = 56, y = 73, rule = B3acik4acjkwy5aery6ei/S2ae3-anq4acikwyz5-acjn6eik7
27b2obo17bo$27b2obobobo11b5o$26b2obobobo11b3ob3o$26bobobobobobobo6b2obob2o$26b2obobobobobo7b3obobo$27b2obobobobobobobo3bobobo$27b3obobobobobobobobobobobobo$28bobobobobobobobobobobobobob2o$28b2o7bobobobobobobobob3o$30bo7bobobobobobobobobo$39bobobobobobobob2o$40bobobobobobob2o$41bobobobobob3o$41b4obobobobo$40b4o3bob4o$40b2o4bob3o13$44bo$8b3o31b5o$6b4o30b4ob4o$4b3obobo7bo19b3obobobob3o$2b4obobob3o5b2o15b4obobobobob4o$2bobobobobob3obobobo14b2obobobobobobobob2o$b3obobobobob3obob3o12b2obobobobobobobobob2o$b2obobobobobobobobob2o12bobobobobobobobobobobo$2obobobobobobobobobob2o10b3obobobobobobobobobobo$b2obobobobobobobob2o2bo10b2obobobobobobobobobo$b3obobobobobobobobob2o10bobobobobobobobobobobo$2bobobobobobob4o2b2o12bobobobobobobobobobo$2b4obobobob2o2b2ob2o11bobobobobobobobobobobo$4b3obobob2obo20bobobobobobobobo$6b4obobo2bo18bobobobobobobobobo$8b3obobobo19bobobobobobobobo$10b4ob2o18bobobobobobobobobo$12bob2o22bobobobobobo$13b2o22bobobobobobobo$12bo25bobobobobobo$39bobobobobobo$10bobo25bobobobobo$7b2o32bobobob2o$7b2obobo25bobobobobo$7bo3bo29bobobobo$8bobob3o25bobobob3o$13b2o26bobobo3b2o$10bob2o28bobobob3o$12bo26bobobobo2bo$13b2o25bobobo2bo$13b2o24bobobobo$13b2o25bobobo$37bobobobob2o$38bobobobob3o$37bobobobobob4o$38bobobobobobobo$36b2obobobobobob3o$36bobobobobobobobo$36b2obobobobob4o$38bobobobob3o$36b2obobobob2o$36b3obobobo$38b4obob2o$40b3ob3o$42b3o!
Checkerboard Expansion
x = 14, y = 14, rule = B2e3aijky4ajrt5ai6c7c8/S1c2-in3nr4actw5aejqr6cen7c8
3bo$3b2o$3b3o$obobo$2ob2o$2bo3$11bo$9b4o$9bob3o$8bobo$9bo$9b2o!
Noodles
x = 31, y = 55, rule = B2e3aijky4acjr5aiy6c7/S1c2-i3knr4a5-ceqy6aci7
19b2o$18b3o$17bo$20bo$17bo$19b2o2$20b2o$20bobo$20b2o6$14b3o$13bo2b2o$
13b2o3$12b3o$14b3o$14bo$13b2o3$11b3o$11b2obo$11b3o11$20b2o$16b2o2bo$
16bobo$14b2ob2o3b2o3b2o$14bobo8bo2b2o$15b2o5bob2ob2o$17bo7b4o$17b2o4b
2o4b2o$27b2obo$23b2o4bo$20bo2b2o2b2o$6b2ob2o12b2o$bobo3b3obo8b2o$o3bo
3bo11bobo$5obo3bo9b2o$8bo!
Reaction Propagation
x = 131, y = 131, rule = B2e3aijky4acjnrty5aci6ce7c/S1c2cek3nr4actw5-ciky6-ek8
48bobo$49bo$48bobo4$58bobo$59bo$58bobo2$42bobo$43bo$42bobo23bobo$69bo$68bobo2$52bobo$53bo$52bobo23bobo$79bo$36bobo39bobo$37bo$36bobo23bobo$63bo$62bobo23bobo$89bo$46bobo39bobo$47bo$46bobo23bobo$73bo$30bobo39bobo23bobo$31bo67bo$30bobo23bobo39bobo$57bo$56bobo23bobo$83bo$40bobo39bobo23bobo$41bo67bo$40bobo23bobo39bobo$67bo$24bobo39bobo23bobo$25bo67bo$24bobo23bobo39bobo23bobo$51bo67bo$50bobo23bobo39bobo$77bo$34bobo39bobo23bobo$35bo67bo$34bobo23bobo39bobo23bobo$61bo67bo$18bobo39bobo23bobo39bobo$19bo67bo$18bobo23bobo39bobo23bobo$45bo67bo$44bobo23bobo39bobo$71bo$28bobo39bobo23bobo$29bo67bo$28bobo23bobo39bobo23bobo$55bo67bo$12bobo39bobo23bobo39bobo$13bo67bo$12bobo23bobo39bobo23bobo$39bo25bo41bo$38bobo23bobo39bobo$63bo3bo$22bobo39bobo23bobo$23bo41bo25bo$22bobo23bobo39bobo23bobo$49bo67bo$6bobo39bobo23bobo39bobo$7bo67bo$6bobo23bobo39bobo23bobo$33bo67bo$32bobo23bobo39bobo$59bo$16bobo39bobo23bobo$17bo67bo$16bobo23bobo39bobo23bobo$43bo67bo$obo39bobo23bobo39bobo$bo67bo$obo23bobo39bobo23bobo$27bo67bo$26bobo23bobo39bobo$53bo$10bobo39bobo23bobo$11bo67bo$10bobo23bobo39bobo23bobo$37bo67bo$36bobo23bobo39bobo$63bo$20bobo39bobo23bobo$21bo67bo$20bobo23bobo39bobo$47bo$46bobo23bobo$73bo$30bobo39bobo23bobo$31bo67bo$30bobo23bobo39bobo$57bo$56bobo23bobo$83bo$40bobo39bobo$41bo$40bobo23bobo$67bo$66bobo23bobo$93bo$50bobo39bobo$51bo$50bobo23bobo$77bo$76bobo2$60bobo$61bo$60bobo23bobo$87bo$86bobo2$70bobo$71bo$70bobo4$80bobo$81bo$80bobo!
Sqrt Growth
x = 10, y = 10, rule = B2e3aijky4acjr5a6cei7c8/S1c2-in3nr4actwy5-ciqy6-ek78
4b2o$4bo$4b2o2$3o4bobo$obo4b3o2$4b2o$5bo$4b2o!
Fractal Fungi
In rule B3aik4acjwy5aeiky6in/S2ae3-anq4acinqwy5-acjn6eik7e 
Zoom
x = 74, y = 74, rule = B2e3aijky4acjr5a6cei7c8/S1c2-in3nr4actwy5-ciqy6aci78
5b2o$7bo$2b2ob2o$2bo$4b3o$obob2obo$obobobo$bo3bo4$11bob2o$12bobo$11bob
o$11b2obo4$20bo$19bobo$18bob2o$19b3o31$52b3o$52b2obo$52bobo$53bo4$59bo
b2o$60bobo$59bobo$59b2obo4$68bo3bo$67bobobobo$66bob2obobo$67b3o$71bo$
67b2ob2o$66bo$67b2o!
