To hopefully complement all the other great posts, here are a few examples where I have seen them used, either directly or lurking around in the background machinery (in their classic or generalised form):

• Studying the elliptic equation $\mathscr{A} u = f$ where $\mathscr{A}$ is the second-order elliptic operator $$(\mathscr{A} u)(x) := - \sum_{i=1}^n \frac{\partial}{\partial x_i} \left(a_i(x) \frac{\partial u}{\partial x_i}\right)(x) + a_0(x) u(x)$$ on some smooth bounded domain $U \subset \mathbb{R}^n$, where either $a_i(x) \to 0$ (i.e. degenerates) or $a_i(x) \to \infty$ (i.e. has a singularity) as $x \to x_0 \in \partial U$. Then we can use a weight to control the blow-up or degeneracy.
• Suppose on the other hand that your elliptic operator $\mathscr{A}$ is "nice" but your data is bad, i.e. consider \left\{\begin{align} - \Delta u &= f, && x \in U,\\u &= g,&& x \in \partial U.\end{align}\right. If $g \in W^{-1/2,2}(\partial U)$ and $f \in W^{-1,2}(U)$ then we can obtain a weak solution $u \in W^{1,2}_0(U)$. However, if $g$ has a singularity at $x_0 \in \partial U$ then $g \notin L^2(\partial U)$ and we cannot use the "standard approach". Introducing a weight that disappears nears $x_0 \in \partial U$ allows us to proceed.
• Suppose $U$ is an unbounded domain. Sometimes when solving boundary value problems, we may want to introduce a weight to specify conditions at infinity. For example, $$\int_{|x|>1} |u(x)|^2(1+|x|)^\epsilon dx < \infty, \quad \epsilon \in \mathbb{R}.$$
• Suppose now that $\partial U$ is no longer a smooth boundary but is quite "nasty", i.e. $\partial U$ might have sharp corners, cusps, etc. Again, when considering a PDE in this domain we can use a weight to control the bad behaviour near the nasty parts of the boundary. This is useful for numerical schemes, such as finite elements.
• Another BVP situation, we might have points on the boundary where the boundary conditions change, for example, from Dirichlet boundary conditions to Neumann boundary conditions. Weights can be used here too.
• When considering the stochastic partial differential equation (SPDE) $$\frac{\partial u}{\partial t} + \Delta u = \dot W(t)$$ where $\dot W$ is a space-time Gaussian white noise, we can generally only obtain solutions in $S'(\mathbb{R}^n)$. As it is nice to obtain function-valued solutions, this can sometimes be achieved if we look for solutions in the weighted space $L^2(\mathbb{R}^n, e^{-\alpha |x|}dx)$.
• Weights also appear in the theory of semilinear equations. For example, for the problem \left\{\begin{align}\frac{\partial u}{\partial t} -\Delta u &= f(u),&& x \in U, t > 0,\\ u &= 0,&& x \in \partial U, t >0,\\ u(t,0) &= u_0(x),&& x \in U\end{align}\right. the concept of a very weak solution may be defined (as a first step to studying blow-up, etc). The definition involves the use of the weighted space $L^1(U,\delta)$ where $\delta(x):=\text{dist}(x,\partial U)$.
• As already mentioned, they appear in Littlewood-Paley type estimates. For example, take the square function $$S^2 f(x_0) = \int_{\Gamma(x_0)} |t \sqrt{\Delta} e^{-t \sqrt{\Delta}} f(x)|^2 \frac{dx dt}{t^n}$$ where $\Gamma(x_0):= \{(x,t):|x-x_0| \le t\}$ with vertex at $x_0 \in \mathbb{R}^{n-1}$ and the nontangential maximal function $u^*(x_0) := \sup\{e^{-t \sqrt{\Delta}}f (x): (x,t) \in \Gamma(x_0)\}$. Then we can show that $\|u^*\|_{L^p(\mathbb{R}^{n-1})}$ and $\|S(f)\|_{L^p(\mathbb{R}^{n-1})}$ are equivalent, and $\|u^*\|_{L^p(\mathbb{R}^{n-1})}$ can be controlled by $\|f\|_{L^p(\mathbb{R}^{n-1})}$. This has implications (again!) for boundary value problems but is also an interesting concept to study on its own.

Finally, in its discrete form:

• I've heard it has applications in the game of Cricket ;-P

Note: I've made this a community wiki. Please feel free to correct any misconceptions, typos, etc.