How a container is read
A comparator facing away from a container reads how full it is. Each slot counts its items against the most that slot can hold, 64, 16 or 1, so a stack of 16 ender pearls fills a slot as much as 64 cobblestone. Those fractions are added up and divided by the number of slots. An empty container gives 0; anything else gives 1 + floor(fullness × 14). So one item in a chest gives 1, and 15 needs every slot full. Checked against Java Edition 26.3; containers work the same way in Bedrock Edition.
The table
The table lists, for one kind of item, the fewest items that give each signal strength and the most before it goes up by one. A hopper of a 64-stack item gives 1 from 1 to 22 items and 2 from 23. Some strengths cannot be reached in small containers: a decorated pot holds one slot, so an item that does not stack takes it straight from 0 to 15. Mix kinds of items in the boxes above to see a reading for what you actually have; it says so when the items need more slots than the container has.
Blocks that are not containers
A few blocks give a signal from their state instead. A composter gives its fill level, a cake two for every slice left, a cauldron its water level, a beehive its honey level, a respawn anchor 0, 3, 7, 11 or 15 by its charges, a lectern how far through the book its page is, a jukebox a number for each disc, a chiseled bookshelf the slot last used, a crafter how many slots are filled or switched off, an item frame how far its item is turned, and an end portal frame 15 with an eye in it. The disc numbers are read from the game's own data for Java Edition 26.3. Bedrock Edition reads most of these the same way, but check a design there before relying on it.
Compare and subtract
A comparator has a rear input and two sides. In compare mode (front torch off) it passes the rear signal on as long as it is at least as strong as the stronger side, and gives 0 otherwise. In subtract mode (front torch on, after a right-click) it gives the rear signal minus the stronger side, never below 0. Both take one redstone tick. Its output then runs along redstone dust losing 1 a block, so a signal of 7 lights 7 blocks of dust.
Questions
How many items for signal strength 1, 2 or 3 in a hopper?
In a hopper of a 64-stack item: 1 item gives 1, 23 give 2 and 46 give 3. For an item sorter's filter hopper the usual setup is 41 of the item to sort plus four slots of one blocker item each, which reads 2 until a 42nd item arrives and makes it 3. Pick Hopper above to see every count.
Why does my comparator read 0 from a full container?
The comparator may be facing the wrong way: it reads the block behind it, the end without the two small torches. A comparator can also read through one solid block between it and the container. Ender chests give no reading at all.
Does the item matter, or only the count?
Only how many and what it stacks to. A slot of 16 snowballs reads the same as a slot of 64 stone, and one sword or potion fills a whole slot.
Does it save anything?
No. Nothing is stored and nothing is sent anywhere. The block and counts are kept in the page's address, so a copied link opens the same reading.