"Least Squares" and "Linear Regression", are they synonyms? What is the difference between least squares and linear regression? Is it the same thing?
 A: Linear regression assumes a linear relationship between the independent and dependent variable. It doesn't tell you how the model is fitted. Least square fitting is simply one of the possibilities.  Other methods for training a linear model is in the comment.
Non-linear least squares is common (https://en.wikipedia.org/wiki/Non-linear_least_squares). For example, the popular Levenberg–Marquardt algorithm solves something like:
$$\hat\beta=\mathop{\textrm{argmin}}_\beta S(\beta)\equiv
\mathop{\textrm{argmin}}_\beta\sum_{i=1}^{m}\left[
  y_i-f(x_i,\beta)
\right]^2$$
It is a least squares optimization but the model is not linear.
They are not the same thing.
A: In addition to the correct answer of @Student T, I want to emphasize that least squares is a potential loss function for an optimization problem, whereas linear regression is an optimization problem. 
Given a certain dataset, linear regression is used to find the best possible linear function, which is explaining the connection between the variables. 
In this case the "best" possible is determined by a loss function, comparing the predicted values of a linear function with the actual values in the dataset. Least Squares is a possible loss function. 
The wikipedia article of least-squares also shows pictures on the right side which show using least squares for other problems than linear regression such as:


*

*conic-fitting 

*fitting quadratic function


The following gif from the wikipedia article shows several different polynomial functions fitted to a dataset using least squares. Only one of them is linear (polynom of 1). This is taken from the german wikipedia article to the topic. 

