What is the norm of a complex number?
In number theory the word "norm" is used with a different meaning than in analysis. While in anaylsis it is important that every positive real number be equal to its own norm, for the use of norm in number theory it is much more important that (for instance) the norm of $a+bi$ be integer if $a$ and $b$ are integers. In some cases the norm may even be negative, for instance in the ring $\mathbf Z[\sqrt 3]$ one would define the norm of $a+b\sqrt 3$ to be $a^2-3b^2$, which is often negative, but it does have the property that an element is invertible if and only if its norm is so (in $\mathbf Z$, i.e., the norm is $\pm1$). In general, in a finite field extension of $K$, the norm of an element is the determinant of the $K$-linear operator defined by multiplication by that element (and the trace of that operation is called the trace of the element).
Let me give you a more geometric perspective of the norm.
If we treat multiplication by $a + bi$ as a linear transformation on the space $x + yi$, then, under the usual basis of $\{1, i \}$, multiplication by $a+bi$ is written as the matrix multiplication:
$$\left(\begin{matrix}a &-b\\b&a\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}ax-by\\ay+bx\end{matrix}\right)$$
In number theory, the "norm" is the determinant of this matrix. In that sense, unlike in analysis, the norm can be thought of as an area rather than a length, because the determinant can be interpreted as an area (or volume in higher dimensions.) However, the area/volume interpretation only gets you so far. The reality is that the determinant of a matrix is an "algebraic" quantity which has the nice property that it is independent of the basis chosen, so it is well defined without picking a basis.
In particular, if $A$ and $B$ are two $n \times n$ matrices, then $\det AB = \det A \det B$, which means that the norm defined this way has the same property: $N(z)N(w)=N(zw)$. That nice property follows through to other cases of "number fields" and their rings of integers where the "area" interpretation is less clear-cut.
In particular, this "algebraic norm" is not measuring distance, but rather measuring something about the multiplicative behavior of $a+bi$. That it turns out to be the square of the geometric norm in this case is a deep geometric fact about the geometry of complex numbers.
Let $z = a+bi \in \mathbb{C}$, where $a$ and $b$ are respectively the real and imaginary parts and $i$ is the imaginary number. Let $z^* = a - bi$ be the conjugate of $z$.
The Euclidean norm ($2$-norm) of $z$ is the defined as
$$\sqrt{zz^*} = \sqrt{(a + bi)(a - bi)} = \sqrt{a^2+b^2}$$
We can define the norm of a complex number in other ways, provided they satisfy the following properties
Positive homogeneity
Triangle inequality
Zero norm iff zero vector
We could define a $3$-norm where you sum up all the components cubed and take the cubic root. The infinite norm simply takes the maximum component's absolute value as the norm. The $1$-norm simply works by taking the sum of the absolute value of all components.
All these norms fulfill the properties above.