How were the Navier-Stokes equations found in the first place if we can't solve them?
I just wanted to give a more concrete idea of how we know these equations even though we have trouble proving analytical theorems about them.
Stuff moving in space
Consider any stuff (as in, any conserved quantity) distributed over space. We know that we can describe this with a time-dependent density field $\rho(x,y,z,t)$ such that any little volume $dV$ has some amount of stuff $\rho~dV$ at that point. We also know that this stuff might be flowing around over time and we formally treat this by saying that we want to know the flow through a little flat surface of area $dA,$ which is oriented in the $\hat n$ direction: that is, the surface is normal to $\hat n$ and "positive" flow will be in the $+\hat n$ direction. Combined together this is a vector $d\mathbf A = \hat n~dA$ and there is some vector field $\mathbf J(x,y,z,t)$ such that the amount of stuff which flows through this area over a time $\delta t$ is $\delta t~d\mathbf A\cdot\mathbf J(x,y,z,t).$ With $\rho$ and $\mathbf J$ we know almost everything. Since the stuff is conserved, we can say that in this box of volume $dV,$ if the amount of stuff in the box changes, it is either because there was a net flow into or out of the sides of the box, so we are doing some $\iint d\mathbf A\cdot \mathbf J$ which turns out by Gauss's theorem to be just $dV~\nabla\cdot\mathbf J,$ or else it came from outside the system we're studying, so there is some term $dV~\Phi$. Equating that to the change in the box $dV~(\partial\rho/\partial t)$ gives the simple starting equation $${\partial \rho\over\partial t} = -\nabla\cdot \mathbf J + \Phi.$$Now when we've got a flow field $\mathbf v(x,y,z,t)$ dictating how a fluid flows, the most dominant transport term is that the box flows downstream, $\mathbf J = \rho~\mathbf v + \mathbf j$ for some deviation $\mathbf j.$ Usually the principal deviation then comes from Fick's law, that there is a flow proportional to the difference in density between adjacent points, $\mathbf j = -D~\nabla \rho,$ but there may be more complex terms there; in particular we shall see pressure here.
Conservation of momentum
The key point here is that $p_x$, the momentum in the $x$-direction, is a stuff. It is a known conserved quantity. It is conserved as a direct result of Newton's third law which turns out, under Emmy Noether's celebrated theorem, to be the same as the statement that the laws of physics are the same at position $x$ as they are at position $x+\delta x$, for a suitable definition of "laws of physics." We are pretty sure about this, and we are pretty sure that the momentum of the fluid itself in the $x$-direction must therefore also be conserved, and this is $\rho~v_x$ where I am shifting definitions a bit on you: $\rho$ now refers to the mass density field and $v_x$ still refers to the fluid velocity in the $x$-direction.
Now a flow of momentum per unit time, which we said is what $\mathbf J\cdot d\mathbf A$ is, is a force. Therefore $\mathbf J$ naturally takes the form of a force per unit area in this context. Now we know that Newton's expression for viscous forces was in fact to write $F_x = \mu~A~v_x/y$ where I am moving a surface of a fluid at speed $v_x$ at a perpendicular distance $y$ from a place where it is being held still; it will not surprise you at all to see that this is very similar to Fick's law and can be written as just $\mathbf j_\text{viscosity} = -\mu~\nabla v_x.$ To that we also need to add the effects of pressure, as a lowering in pressure also drives a fluid motion; this is a little bit harder to reason out but it takes the form that we can imagine a constant flow in the $x$-direction of $p~\hat x$ and then deviations in this flow would produce the change in momentum per unit time $-\partial p/\partial x$ through this divergence term. (That's a little bit of a sloppy way to show that we are talking about a stress tensor and part of it is $p~\mathbf 1$, the identity matrix multiplied by the pressure.) Combining these two components of $\mathbf j$ we have $${\partial \over\partial t}(\rho~v_x) = -\nabla\cdot (\rho~v_x~\mathbf v - \mu \nabla (v_x)) - \frac{\partial p}{\partial x} + \Phi_x.$$The external contribution $\Phi$ comes from forces influencing the fluid from outside, like gravity.
In the Navier-Stokes equations the Millenium Prize has restricted itself to a considerably simpler case where $\nabla\cdot\mathbf v = 0$ and $\rho$ and $\mu$ are constant, which we call "incompressible flow." This is generally a valid assumption when you're interacting with a fluid at speeds much lower than the speed of sound in that fluid; then the fluid would rather move away from you than be compressed into any one place. In this case we can commute $\rho$ out of all of the spatial derivatives and then divide by it, so that the only impact is to rewrite $\nu=\mu/\rho$ and $\lambda=p/\rho$ and $a_x=\Phi_x/\rho$, eliminating the unit of mass from the equation. For $v_x$ we have specifically, $${\partial v_x\over\partial t} + \mathbf v\cdot\nabla v_x - \nu \nabla^2 v_x = - \frac{\partial \lambda}{\partial x} + a_x,$$ and then we can extend the above analysis to the directions $y,z$ too to find, $$\dot{\mathbf v} + (\mathbf v\cdot\nabla)\mathbf v - \nu \nabla^2 \mathbf v = - \nabla \lambda + \mathbf a.$$This is the version of the Navier-Stokes equations written down in the Millenium Prize; we have a very straightforward explanation of this as "The flow of momentum in a small box flowing downstream in an incompressible homogeneous Newtonian fluid is due entirely to Fick's-law diffusion of the momentum due to the viscosity of the fluid, plus a force due to pressure gradients inside the fluid, plus forces imposed by the external world."
Why this equation?
The understanding of the physics of how we got to this equation is not in question. What's at stake is the mathematics of this equation, in particular this $(\mathbf v \cdot \nabla) \mathbf v$ term which contains $\mathbf v$ twice and thereby makes it a nonlinear partial differential equation: given two flow fields $\mathbf v_{1,2}$ which are valid, in general $\alpha \mathbf v_1 + \beta \mathbf v_2$ will not solve this equation, removing our most powerful tool from our toolbox.
Nonlinearity turns out to be unbelievably hard to solve in general, and essentially the Clay Mathematics institute is giving the million-dollar prize for anyone who cracks nonlinear differential equation theory strongly enough that they can answer one of the more basic mathematical questions about these Navier-Stokes equations, as a "most basic example" for their new theoretical toolkit.
The idea of the Clay prizes is that they are specific problems (which is important for awarding a prize for their solution!) but that they seem to require powerful new general ideas which would allow our mathematics to go into places where it has historically been unable to go. You see this for example in $\text{P} = \text{NP}$, it's a very specific question but to answer it we would seem to need to have a better handle on "here's a classification of set of stuff which computers can do, and here are some things which a computer can't efficiently do" which nobody has yet been able to convincingly present. A new toolbox which could resolve this "stupid little" question would therefore profoundly improve our ability to work on a huge class of related problems in computation.
As @QMechanic mentioned in a comment, the Navier-Stokes equations are just $F = ma$, but they look much scarier. Assuming an incompressible fluid, you have:
$$ \rho \frac{D u_i}{D t} = -\frac{\partial \sigma_{ij}}{\partial x_j} + f_b $$
where $\rho$ is the density (mass per unit volume), $D u_i /D t$ is the acceleration (written in the Lagrangian form, rather than Eulerian for clarity), $\sigma$ is the Cauchy stress tensor (generally expanded to take out the pressure from the trace, $\sigma_{ij} = -p \delta_{ij} + \tau_{ij}$ where now $\tau_{ij}$ is the viscous stress tensor), and $f_b$ is the body force (things like gravity).
Everything on the right hand side is the sum of forces, and the left is the mass times the acceleration.
The Navier Stokes equations are a combination of Newton's 2nd law of motion (differential form) with the 3D version of Newton's law of viscosity (i.e., the mechanical constitutive equation for a Newtonian fluid).
What you were taught in Physics at school was correct. Not many analytic solutions exist to interesting practical problems, and numerical solution (say using computational fluid dynamics (CFD)) is often necessary. But they are certainly well understood.