Consider the set $X=\{1,2,3\}$.
- We have the trivial topology, namely $T=\{\emptyset,\{1,2,3\}\}$.
- We have the discrete topology, in which every singleton is open. This produces the topology to be $P(X)=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$
- We may have something in between declaring the nonempty open sets to be those containing $1$, and then we have $\{\emptyset,\{1\},\{1,2\},\{1,3\},\{1,2,3\}\}$.
Closed sets are those whose complement is open, and clopen sets are those which are open and closed as well.
Note that $X$ is always clopen relatively to $X$ (it might not be if we take a larger set and endow it with a different topology).
In the trivial topology we only have clopen sets, because we only require the minimum from the topology, while the set $\{1,2\}$ is neither open nor closed.
In the discrete topology every subset of $\{1,2,3\}$ is open, therefore every subset is closed. Since every subset is both open and closed, every subset is clopen.
In the last example note that all those that include $1$ are open, so those who do not include $1$ are closed. Since $1$ cannot be in both a set an its complement we have that there are no clopen sets except the empty set and $\{1,2,3\}$.
However it is important to remember that this is all relative and clopen, open and closed sets are only in this relationship with a specific topology and space.
Addendum: I feel that I should add a very basic explanation about topology.
Suppose we have an underlying set $X$. A topology on $X$ is a collection of subsets which includes the empty set as well $X$ itself. It is closed under unions and finite intersections.
The sets which are in the topology are called open, and their complements are called closed. A set which is both open and closed is usually called clopen.