What is the difference between a one dimensional and a two dimensional signal?
Basically the dimensional number refers to the number of independent variables (input). 

In one dimensional signal $f(t)$, the amplitude $y=f(t)$ is the dependent variable (output), and there is only one independent variable $t$. 

In two dimensional image $f(x,y)$, the two independent variables are $x$ and $y$, and the dependent variable $f$ is also called intensity. If you want to make it similar with amplitude vs time plot as you do with one dimensional signal, you can plot a intensity vs x vs y scattered dots with three axis as well.