If you accept that arithmetic (addition and multiplication) is interesting (and fundamental to mathematics) then polynomials are a simple way to write down rules for processes of addition and multiplication.
Then one can observe that there are simple rules for manipulating these processes, which amounts to doing algebra with polynomials. Moreover, there is lots of structure inside the processes themselves (factorization, derivatives, Galois theory, algebraic geometry, representation theory, invariant theory, etc.).
Let's return to the idea of polynomials as description of a process for doing arithmetic with numbers.
For example, $x^2 + 1$ expresses the process of squaring a number, and adding one. For real $x$ , this expression never less than 1, so has no roots. However, the catch was in the description of a polynomial as "a process for doing arithmetic with "numbers.""
What numbers are we considering? We could have plugged in only whole numbers instead of the reals and features of this process would be different (e.g. the output would be whole numbers, and have "jumps"). (So perhaps there are some things that are intrinsic to the process, and some things that depend on what numbers you are "plugging in" to it. This point of view turns out to be very fruitful ... for more on this, see the article by Jordan Ellenberg in the Princeton Guide to Pure Mathematics: http://press.princeton.edu/chapters/gowers/gowers_IV_5.pdf )
As you (may) know, we can construct the complex numbers, where the "process" $x^2 + 1$ can result in $0$, by plugging in $i$ or $-i$.
Moreover, one can build the complex numbers using polynomials with real coefficients. The key observation is that, like numbers, you can add and multiply polynomials, and you can also multiply polynomials by a real number. So given any polynomial $p(x)$, you can plug a polynomial into it to obtain a new polynomial.
If I declare the polynomial $t^2 + 1$ to be $0$ (just go with it for now), then when I plug $t$ into $x^2 + 1$, I get $t^2 + 1$, which I have said is zero. So, if I study all polynomials of the form $\Sigma a_i t$,with $a_i$ real, I appear to have a solution to the equation $x^2 + 1$. (This means that I have a number,"$t$", that I can input into the process $x^2 + 1$ a.k.a "square and add one", and get out the thing which I call $0$.)
So $t$ behaves like $i$, and indeed one can prove that this system of polynomials has all the properties that we want the complex numbers to have. From an operational point of view (the properties we want some object to have), we have constructed the complex numbers out of polynomials and real numbers.
For (much) more on this, you can try to read Ian Stewart's Galois Theory. (It will be hard, but you shouldn't be afraid. If I recall correctly, it doesn't assume much by way of prerequisites, but you may need to consult some other texts on abstract algebra along the way.)
Lastly -- for a visual point of view on polynomials, I find it helpful (in some situations, certainly not for computation), to identify polynomials with their roots. I can then imagine the roots floating around on the real line (or complex plane). (Note that this point of view does not distinguish $x^2$ and $2x^2$, but does distinguish $x^2$ from $x$, if you imagine that there are two points at zero when visualizing $x^2$.) This leads to the notion of configuration space - the "space" of points in some other "space." (And also has connections to Galois theory, via "covering" spaces, when you try to imagine what happens to the roots as we move the coefficients of the polynomials around... for example, try this for $z^2 - a$, as you move $a$ clockwise on the unit circle in the complex plane.)