I saw in a recent answer a reference to 1/3 casters and 1/2 casters, and have never encountered this terminology before:

Paladin's are known as 1/2 casters, so their spell progression isn't as big as full casters.

What does it mean to be a 1/3 caster vs a 1/2 caster vs a full caster? How does one distinguish between the three, and which classes correspond to which types of caster?

It refers to the speed at which classes gain spell slots and new levels of spells during their progression. Full casters (like a Wizard) gain spell slots fastest, while 1/3rd casters only gain slots and new spell levels at about a third that rate.

The name "1/2 caster" and "1/3 caster" comes from the Multiclassing rules, which state that if you have levels in multiple classes capable of casting spells, you use a specific table to determine your spell slots. You only get to add half and a third of the levels in those classes (rounded down) to determine your total spell slots.

You can read more about this in the PHB on page 164, under the header "Spellcasting".

For a list of classes:

Full casters

• Bard


• Cleric


• Druid


• Sorceror


• Wizard

Half casters

• Paladin


• Ranger

Third casters

• Eldritch Knight Fighters


• Arcane Trickster Rogues

Special cases

Warlocks are a special case; their spellcasting works very differently. They are explicitly called out in the multiclass rules as not providing extra base slots at all.

"1/X Caster" is shorthand for how quickly a character gains newer, more powerful spell slots

The progression for each of these spellcaster types looks like this, with the first column representing Character Level as a Single-classed X and the other columns representing the "Spellcaster Level" they have as that Single-classed X at a given level.

begin{array}{l|l|l|l}
textbf{Character Level} & textbf{Full} & textbf{Half} & textbf{Third} \ hline
1 & 1 & - & - \
2 & 2 & 1 & - \
3 & 3 & 2 & 1 \
4 & 4 & 2 & 2 \
5 & 5 & 3 & 2 \
6 & 6 & 3 & 2 \
7 & 7 & 4 & 3 \
8 & 8 & 4 & 3 \
9 & 9 & 5 & 3 \
10 & 10 & 5 & 4 \
11 & 11 & 6 & 4 \
12 & 12 & 6 & 4 \
13 & 13 & 7 & 5 \
14 & 14 & 7 & 5 \
15 & 15 & 8 & 5 \
16 & 16 & 8 & 6 \
17 & 17 & 9 & 6 \
18 & 18 & 9 & 6 \
19 & 19 & 10 & 7 \
20 & 20 & 10 & 7 \
end{array}

The classes that fall into these categories are:

begin{array}{l|l|l|l}
textbf{Full} & textbf{Half} & textbf{Third} & textbf{Other}\ hline
text{Bard} & text{Artificer* (new)} & text{Arcane Trickster} & text{Warlock†} \
text{Cleric} & text{Paladin} & text{Artificer (old)} \
text{Druid} & text{Ranger} & text{Eldritch Knight} \
text{Sorcerer} \
text{Wizard} \
end{array}

^{*The Revised Artificer is a Half-Spellcaster, but unlike other Half Spellcasters, they gain their spellcasting feature at level 1, instead of level 2, and are treated like level 1 spellcasters at that level.
†Warlocks are unique in that their spell access resembles that of a Full Spellcaster, but their Spell Slots are completely divorced from the system that all other spellcasters use, so they need their own category.}

So if you're a level 9 Paladin (Half Spellcaster), you have spell slots equivalent to a level 5 Cleric (Full Spellcaster) and to a level 13 (or 14 or 15) Eldritch Knight (Third Spellcaster)—and for each of these characters, their Spell Slot total is 4 1st Level Slots, 3 2nd Level Slots, and 2 3rd Level Slots.

"1/X Caster" also comes into play in the Multiclassing Rules

What kind of Spellcaster you are affects how your levels are added together when you Multiclass into multiple kinds of spellcaster.

Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature.† Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.

—Multiclassing, Player's Handbook, pg. 164

^{†Artificers have a special rule: when adding their levels for multiclassing purposes, you round up after dividing by two, instead of rounding down. Note also that Warlocks are not included in this list; again, their Spellcasting is completely different from other classes, so they aren't considered in calculating a character's normal spellcasting level.}

For example, suppose we have a Multiclassed Wizard 5/Eldritch Knight 11. We add their levels by first dividing them by the level of spellcaster they are, so we take 5 Wizard Levels (5 * 1/1 = 5) and 11 Eldritch Knight Levels (11 * 1/3 = 3.666 → Rounded Down to 3) and add them together to find that this character is the equivalent of a Level 8 Spellcaster, gaining 4 1st Level Slots, 3 2nd Level Slots, 3 3rd Level Slots, and 2 4th Level Slots.

The fractions here come from the multiclassing rules: when figuring out what level of “multiclassed spellcaster” you are, in order to figure out your spell slots, you need to add your full level in some classes (like wizard), half your level in other classes (like paladin), and a third of your level in yet other classes (like arcane trickster). These fractions roughly correspond to how much spellcasting those classes actually give: wizards and other “1” classes get 9th-level spells, paladins and other “½” classes get 5th-level spells, and arcane tricksters and other “⅓” classes get 4th-level spells.