How does replacing x with x-a shift a curved line to the right?

My professor just took up notes regarding curved lines and surfaces. In the curved lines section, he said a curved line on the 2D plane is an equation with x and y. He followed by saying if we replace x with x-a, the graph is shifted to the right by a. So, if we look at y = 2x.

x| 0 1 2 3 4
-------------
y| 0 2 4 6 8


Let a = 1.

x-1| -1 0 1 2 3
---------------
y  | -2 0 2 4 6


The graph simply moved down and left by 1 (a).

If we look at y = x^2.

x| 0 1 2 3
----------
y| 0 1 4 9


Let a = 1.

x-1| -1 0 1 2
-------------
y  |  1 0 1 4


The graph just shrunk by 1 (a). What does he mean when he says "if we replace x with x-a, the graph is shifted to the right by a"?

You appear to have misunderstood what your professor told you. Taking your first example of $$y=2x$$, we replace $$x$$ by $$x-a$$ in the equation to get the new equation $$y=2(x-a)=2x-2a$$. If you plot the resulting graph for $$a=1$$, you’ll see that it is indeed the original line shifted one unit to the right. Similarly, substituting into $$y=x^2$$ produces $$y=(x-1)^2=x^2-2x+1$$. If you plot this, you’ll see that it’s the original parabola shifted one unit to the right.

This is not completely wrong. The remaining to recall how to plot a graph: to find out what $$y$$ is for each $$x$$.

For your first example, translating $$y=2x$$ to the right by replacing $$x$$ with $$x-1$$:

$$\begin{array}{c|c|c|c|c} x-1 & -1 & 0 & 1&2&3\\\hline y = 2(x-1) & -2&0&2&4&6\\\hline \color{red}x & 0 & 1&2&3&4 \end{array}$$

When you plot the $$y$$ values for each $$\color{red}x$$, the graph is indeed translated $$1$$ unit to the right. (e.g. the root is now $$x=1$$)

Another example, translating $$y = x^2$$ to the right by $$1$$ unit:

$$\begin{array}{c|c|c|c} x-1 & -1 & 0 & 1&2\\\hline y = (x-1)^2 & 1&0&1&4\\\hline \color{red}x & 0 & 1&2&3 \end{array}$$

Now try to plot the $$xy$$ graph.