L2 loss vs. mean squared loss
Function $L_2(x):=\left \|x \right \|_2$ is a norm, it is not a loss by itself. It is called a "loss" when it is used in a loss function to measure a distance between two vectors, $\left \| y_1 - y_2 \right \|^2_2$, or to measure the size of a vector, $\left \| \theta \right \|^2_2$. This goes with a loss minimization that tries to bring these quantities to the "least" possible value.

These are some illustrations:


$L_p$ norm: $L_p(x) := \left \|x \right \|_p = (\sum_{i=1}^{D} |x_i|^p)^{1/p}$,
where $D$ is the dimension of vector $x$,
Squared error: $\mbox{SE}(A, \theta) =\sum_{n=1}^{N} \left \| y_n - f_{\theta}(x_n) \right \|^2_2$,
where $A=\{(x_n, y_n)_{n=1}^{N}\}$ is a set of data points, and $f_{\theta}(x_n)$ is model's estimation of $y_n$,
Mean squared error: $\mbox{MSE}(A, \theta) =\mbox{SE}(A, \theta)/N$,   
Least squares optimization: $\theta^*=\mbox{argmin}_{\theta} \mbox{MSE}(A, \theta)$$=\mbox{argmin}_{\theta} \mbox{SE}(A, \theta)$,  
Ridge loss: $\mbox{R}(A, \theta, \lambda) = \mbox{MSE}(A, \theta) + \lambda\left \| \theta \right \|^2_2$
Ridge optimization (regression): $\theta^*=\mbox{argmin}_{\theta} \mbox{R}(A, \theta, \lambda)$.


In all of the above examples, $L_2$ norm can be replaced with $L_1$ norm or $L_\infty$ norm, etc.. However the names "squared error", "least squares",
 and "Ridge" are reserved for $L_2$ norm. For example for $L_1$, "squared error" becomes "absolute error":


Absolute error: $\mbox{AE}(A, \theta) =\sum_{n=1}^{N} \left \| y_n - f_{\theta}(x_n) \right \|_1$,