Although these inequalities occur in various settings, and I have used them to complete a number of proofs, I can not say that I intuitively understand what their significance is.
Holder's Inequality:
Given $p,q > 1$ and $\frac{1}{p} + \frac{1}{q} = 1$, and $(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{R}^n\text{ or }\mathbb{C}^n.$ $$ \sum_{k=1}^n |x_k\,y_k| \leqslant \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\!1/p\;} \biggl( \sum_{k=1}^n |y_k|^q \biggr)^{\!1/q} $$
Cauchy-Schwarz Inequality:
$$ \left(\sum_{k = 1}^n x_ky_k\right)^2 \leqslant \left(\sum_{k = 1}^nx_k^2\right)\left(\sum_{k = 1}^n y_k^2\right) $$
I have noticed that they tend to come up in situations where symmetry between the vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ is of interest. It seems to me that equality is approached as these vectors approach being parallel; is this correct?
I'd appreciate any insight into what these inequalities are measuring.