What is an algebraic group, and what four properties must it satisfy?

Question

Accepted Answer

A group $\langle G, * angle$ is a set G with an operation * that satisfies closure, associativity, has a neutral element, and every element has an inverse.
The four group axioms:

Property
Requirement

Closure
$a * b \in G$ for all $a, b \in G$

Associativity
$a * (b * c) = (a * b) * c$

Neutral element
There exists $e \in G$ such that $a * e = e * a = a$

Inverse element
For every $a \in G$, there exists $a' \in G$ with $a * a' = a' * a = e$

If additionally commutativity holds ($a * b = b * a$), it's called an Abelian group (after Niels Henrik Abel).
$|G|$ = Order of the group = number of elements.
Examples of groups:

Group
Operation
Neutral
Inverse of $a$
Order

$\langle \mathbb{Z}_n, + \mod n angle$
Addition
0
$-a \mod n$
$n$

$\langle \mathbb{Z}_p^*, \cdot \mod p angle$ ($p$ prime)
Multiplication
1
$a^{-1} \mod p$
$p - 1$

$\langle \mathbb{Z}_n^*, \cdot \mod n angle$
Multiplication
1
$a^{-1} \mod n$
$\varphi(n)$

Not groups: $\langle \mathbb{N}, + angle$ (no inverse: $-a 
otin \mathbb{N}$), $\langle \mathbb{Z} \setminus \{0\}, \cdot angle$ (no inverse: $a^{-1} 
otin \mathbb{Z}$)
Go deeper:

Group (mathematics) (Wikipedia) — the four axioms with worked examples.
 The definition of a Group (Socratica) — a crisp 6-minute walk through the axioms.

Property	Requirement
Closure	$a * b \in G$ for all $a, b \in G$
Associativity	$a * (b * c) = (a * b) * c$
Neutral element	There exists $e \in G$ such that $a * e = e * a = a$
Inverse element	For every $a \in G$, there exists $a' \in G$ with $a * a' = a' * a = e$

Group	Operation	Neutral	Inverse of $a$	Order
$\langle \mathbb{Z}_n, + \mod n \rangle$	Addition	0	$-a \mod n$	$n$
$\langle \mathbb{Z}_p^*, \cdot \mod p \rangle$ ($p$ prime)	Multiplication	1	$a^{-1} \mod p$	$p - 1$
$\langle \mathbb{Z}_n^*, \cdot \mod n \rangle$	Multiplication	1	$a^{-1} \mod n$	$\varphi(n)$