Definition:
A function $f$ is continuous at a number $x=a$ if
$\displaystyle \lim_{x \rightarrow a} f(x) = f(a)$.

Remember that $\displaystyle{\lim_{x \to a} f(x)}$ describes both
what is happening when $x$ is slightly less than $a$ and what is
happening when $x$ is slightly greater than $a$. Thus there
are three conditions inherent
in this definition of continuity. A function is continuous at $a$ if the limit as
$x\to a$ exists, and $f(a)$ exists, and this limit is equal to
$f(a)$. This means that the following three values are
equal: $$\lim_{x \to a^-} f(x)\qquad=\qquad
f(a)\qquad=\qquad\lim_{x \to a^+} f(x)$$ I.e. the value as
$x$ approaches $a$ from the left is the same as the value as $x$
approaches $a$ from the left (the limit exists) which is the same
as the value of $f$ at $a$.

If any of these quantities is different, or if any of them fails to
exist, then we say that $f(x)$ is discontinuous
at $x=a$, or that $f(x)$ has a discontinuity
at $x=a$.

What does this mean graphically? If you trace $f$ with a
pencil from left to right, as you approach $x=a$, you are at some
height $L$ (because $\displaystyle\lim_{x\to a^-}f(x)=L$. As
you go through the $x$-value $a$, your height is also $L$ (because
$f(a)=L$). Now as you keep going with your pencil, you are
beginning this last stretch at height $L$ (because
$\displaystyle\lim_{x\to a^+}f(x)=L$).

DO: Sketch $f(x)=\sqrt x$
and let $a=4$. Find $f(a), \displaystyle\lim_{x\to
a^-}f(x)=L$, and $\displaystyle\lim_{x\to a^+}f(x)$. Now,
follow along your graph as stated in the previous paragraph,
looking at each condition as you go. Is $f(x)=\sqrt x$
continuous at $x=a$?

Now, more interestingly, consider $\displaystyle\frac{x^2-1}{x-1}$
in this video:

Definition:
A function $f$ is
continuous from the right at $x=a$ if $\displaystyle
\lim_{x \rightarrow a^+} f(x) = f(a)$, and is continuous from the left at $x=a$ if
$\displaystyle \lim_{x \rightarrow a^-} f(x) = f(a)$, and is
continuous on an interval $I$ if it is
continuous at each interior point of $I$, is continuous from
the right at the left endpoint (if $I$ has one), and is
continuous from the left at the right endpoint (if $I$ has
one).