In first-order logic we are allowed to put a quantifier only on first-order variables. This means that when we interpret the language and ask whether or not a sentence is true or false, then we can only quantifier on elements of the universe. In second-order logic, however, we are allowed to quantify over relations, or subsets of the universe as well.
For example, the statement "Every bounded set of real numbers has a least upper bound" is a second-order statement. We quantified over all the sets of real numbers, and we say "If $A$ is a bounded set, then $\sup A$ exists".
On the other hand, "$\sqrt2$ exists" is a first-order statement (in the language containing $+,\cdot,0,1$, that is) because we can characterize $\sqrt2$ as an element $x$ such that $x\cdot x=1+1$.
Second-order logic, if so is a much stronger way to write things. It allows us to express a lot more in comparison to first-order logic.
So why are we mostly interested in first-order logic? First of all, there is a lot of research into other strong logics (logics which allow us to express more than first-order itself, but usually less than second-order logic). Secondly, first-order logic got the completeness theorem and the compactness theorem, as well the Lowenheim-Skolem theorems. These mean that we have a great model theory with first-order logic. On the other hand, we don't have those for stronger logics (either we don't have one of them, or we don't have both) which means that proof theory and model theory are going to be harder to work with.
Moreover second-order logic has several variants. For example we may allow only to quantify over subsets of the universe, not general relations; or we may interpret the second-order logic in a way that the subsets we are interested in were all definable by a first-order formula (Henkin semantics). These are weaker than just full-on quantification and so on. The Henkin semantics version of second-order logic is equivalent to first-order logic in a very good sense.
I feel that I'm already confusing you enough. So let's stop here.