What does it mean for a function to be continuous on its domain? I never understood the phrase "continuous on its domain."

Isn't everything continuous on its own domain, since the domain are all the $x$ values that we can plug into $f(x)$ and get a defined $y$ value back? i.e. doesn't the domain by definition tell you where the function is continuous? Why would the domain ever include something not continuous / not defined?
Not necessarily. The domain of a function tells you over what values the function $f(x)$ exists, not where it is continuous. Take the piecewise function:

$$f(x) = 
\begin{cases}
1 &amp; x&lt;0\\
2 &amp; x\geq0
\end{cases}$$

This function is defined for all $x\in\mathbb{R}$, but is not continuous at $x=0$. It still has a valid value: $f(0)=2$, but that doesn't make it continuous at that point.

For a function to be continuous at a point, its limit must be the same regardless of what direction of approach. In this case, $\lim\limits_{x\to0^-}{f(x)}=1$ while $\lim\limits_{x\to0^+}{f(x)}=2$, making it discontinuous at that point.
A function is said to be continuous if it continues at each point. This means that over the domain. Functions that are not continuous do not exist for every x value over the domain. For example if a function is defined near an open interval (the circle that is not shaded on a graph) then the function is discontinuous. However, if a function is defined near a closed circle (the shaded circle of a graph) then that function is continuous. This problem usually occurs when trying to find the Domain and Range of a function or if a problem ask to graph the Domain and Range of a function.