Loss function for regression

As said in the first comment, the first version is one to evaluate empirically on given data, whereas the second one with $E[L]$ is a theoretical population version for a general loss function $L$; in the first equation $L(t,y(x))=(t-y(x))^2$, but the second one can use other loss functions if desired (which will then have empirical versions as well, summing up other losses).

The given formula for $E[L]$ assumes both $x$ and $t$ to be random (sometimes in regression modeling $x$ is assumed to be fixed, but it's probably not worthwhile to go into this because it doesn't make that much of a difference for the question). Regarding the distribution $p(x,t)$, obviously for evaluating the theoretical $E[L]$ one needs to make some model assumptions about it, however in empirical data the best that we have is the empirical distribution of the $t$ and the $x$. Now if we evaluate the integral in $E[L]$ with $p$ being the empirical distribution of $(x,t)$ on a given dataset of size $N$ (i.e., every observed $(x_i,t_i)$ appears with probability $frac{1}{N}$), it is actually $frac{1}{N}sum_{i=1}^NL(t_i,y(x_i))$, and with the squared loss $frac{1}{N}sum_{i=1}^N(t_i-y(x_i))^2$ (if we consider $N$ as fixed, the factor $frac{1}{N}$ is just a constant that doesn't matter). This connects the two formulae. (When making model assumptions about $p$ in order to say something theoretical about $E[L]$, one would hope that they are more or less in line with the empirical distribution of a dataset to which theoretical results are applied.)