What does a tilde underneath an inequality mean? I've recently come across an expression of the form
$$\large x \lesssim y$$
What does this expression mean?

In the context of partial differential equations and harmonic analysis, this notation is often used to mean 
$$ A \lesssim B \iff \exists C > 0 \text{ s.t. } A \leq C B $$
As written this expression is pretty damn useless when $A$ and $B$ are just two real numbers. What's more useful is supposing $A(\lambda), B(\lambda)$ are two families of objects parametrised by $\lambda\in \Lambda$ for some set. Then 
$$ A \lesssim B \iff \exists C > 0 \text{ s.t. } \forall \lambda\in \Lambda~,~ A(\lambda) \leq C B(\lambda) $$
Sometimes you will see when $A(\lambda,\pi), B(\lambda,\pi)$ are two families of objects parametrised by $(\lambda,\pi) \in \Lambda\times\Pi$, the notation 
$$ A\lesssim_\pi B \iff \forall \pi\in \Pi \exists C = C(\pi)>0 \text{ s.t. } \forall \lambda\in \Lambda ~,~ A(\lambda,\pi) \leq C(\pi) B(\lambda,\pi) $$
that is, the constant in the inequality is universal over $\lambda$ but may depend on $\pi$. 

Similarly one sees the notation 
$$ A\approx B \iff A\lesssim B \text{ and } B\lesssim A $$
which also has the $\approx_\pi$ variant in the same way. 
