What's the difference between $∀x\,∃y\,L(x, y)$ and $∃y\,∀x\,L(x, y)$? 
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*Everybody loves somebody. $∀x\,∃y\,L(x, y)$

*There is somebody whom everybody loves. $∃y\,∀x\,L(x, y)$
What's the difference between these two sentences? If they are same, can I switch $\exists y$ and $\forall x$?

If $L$ satisfies 2., then it necessarily satisfies 1. Therefore you can switch $\exists y$ and $\forall x$ to go from 2. to 1., but not the other way around. Counterexample: let $L$ be a relation over set $S=\{a,b,c\}$, and suppose $L(a,b)$, $L(b,c)$, $L(c,a)$. You can easily verify that 1. holds here, but 2. does not.
