Regular Expressions to Automaton
There are (at least) three algorithms that convert a regular expression to an equivalent finite-state automaton:


Thompson's construction (see e.g. the Wikipedia article)
McNaughton and Yamada's construction
Glushkov's construction (see e.g. the Wikipedia article)


Büchi automata enter the picture if you are working with $\omega$-regular expressions. See e.g. this description of the algorithm for constructing an equivalent Büchi automaton from an $\omega$-regular expression.