the power set
The power set of a set A is the set of all of A's subsets — every possible way to pick zero or more of its elements. Imagine you have a small wardrobe and you list every possible outfit, including the 'wear nothing from this wardrobe' choice and the 'wear all of it' choice: that complete catalogue of selections is the power set. It is a set whose members are themselves sets, which feels strange at first but is perfectly ordinary.
We write the power set of A as P(A) or 2^A. If A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}}: the empty selection, the two single-element selections, and the whole thing. A neat counting fact explains the 2^A notation: for each of the n elements you make an independent yes/no choice about whether to include it, giving 2 × 2 × ... × 2 = 2^n subsets in total. So a 3-element set has 2^3 = 8 subsets, and even a 10-element set already has 1024.
The power set is the secret behind one of the most important constructions in this whole subject: turning a nondeterministic finite automaton into a deterministic one. Because an NFA can be 'in several states at once', a DFA simulating it must track which subset of the NFA's states is currently active — so the DFA's states are exactly the members of the power set of the NFA's states. This is also why that conversion can cause an exponential blowup: n NFA states can in principle produce up to 2^n DFA states.
P({1, 2, 3}) has 2^3 = 8 members: ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. Note that both ∅ and the whole set {1,2,3} are always members of the power set.
A 3-element set has 8 subsets, two of which are the empty set and the whole set.
Do not confuse an element with a one-element subset: a ∈ A is not the same statement as {a} ⊆ A. The members of P(A) are sets, so {a} ∈ P(A), but a ∉ P(A).