What is a "lattice" in set theory???

To understand lattices first you need to understand partially ordered sets. A partially ordered set is a set with an ordering operation $ \leq $ that sometimes works.

For example the set of all people ordered by ancestry is a partially ordered set (poset). If your mother is your ancestor and your mother's mother is your ancestor then your grandmother is your ancestor. However you are neither your siblings ancestor nor is your sibling your ancestor.

A lattice is a poset with two additional restrictions:

For any two members $x,y$ of the set there is a member of the set which is larger than or equal to both $x$ and $y$, and is the smallest member that has this property. This is called their join, and is denoted $x \vee y$.

The other restriction is that for any two members $x,y$ of the set there is a member of the set which is smaller than or equal to both $x$ and $y$, and is the largest member that has this property. This is called their meet, and is denoted $ x\wedge y $.

A lattice is an algebraic structure, generalizing each of the following pairs of (binary) operations: $$\min,\ \max$$ $$\inf,\ \sup$$ $$\bigcap,\ \bigcup$$ $$\mathtt{and},\ \mathtt{or}$$ $$\gcd,\ \mathrm{lcm}$$

A partially ordered set can be naturally equipped with an algebraic lattice structure whenever every pair of elements $a,b$ has a greatest lower bound $a\land b$, and a least upper bound $a\lor b$.

(Conversely, every lattice determines a partial order by $a\le b\ :\iff\ a\lor b=b \ (\iff a=a\land b)$.)

Consider a partially ordered set $(A,\leq)$, and a non-empty subset $B$. A lower bound is an element $l$ such that for every $b \in B, l \leq b$. A greatest lower bound is an element $l_0$ such that for every lower bound $l$, $l \leq l_0$.

An example would be $A = \mathbb{R}$ with the standard order, $B$ the subset $(0,1) \cup (2,3)$. Any element less than or equal to $0$ is a lower bound, and the greatest lower bound is $0$.

An example of a subset without a greatest lower bound would be $A = \mathbb{R}, B = (\infty, 1]$. In fact $B$ has no lower bound at all.

Similar definitions apply for upper bounds and greatest upper bounds.

A lattice is a partially ordered $(A, \leq)$ with the property that for every $a,b \in A$, the set ${a,b}$ has a greatest lower bound and a least upper bound.

An example of a lattice would be $A = [0,1]$ with the standard ordering. An example of something that is not a lattice would $A = \{\{1\},\{2\},\{1,2,3\},\{1,2,10\}\}$ with set theoretic inclusion as the ordering. Taking $a = \{1\}, b = \{2\}$ the set $B = \{a,b\}$ has two upper bounds $\{1,2,3\}$ and $\{1,2,10\}$ but does not have a greatest upper bound, since $\{1,2,3\}$ and $\{1,2,10\}$ are not comparable. (Note that they are not comparable because neither is a superset of the other)

Another example of a lattice would be the powers of a set with set theoretic inclusion.

A way to think of lattices would be as a sort of structure where every pair of elements has one element above it that is smaller than every other element above it, and one bigger then every below it. Visualized this means that every pair of elements forms either has one element above and one below, or forms a diamond with some pair of elements, one above and one below.