Thompson construction
Thompson construction builds an ε-NFA from a regular expression by walking the expression tree and wiring small gadgets for each operator.
Each piece has one entrance and one exit. Union, concatenation, and Kleene star always connect with ε-edges the same way. The result is usually fatter than a hand-drawn NFA; ε is glue so the translation is mechanical and easy to prove correct.
After Thompson you remove ε-moves (ε-closure → ordinary NFA), then optionally subset-construct a DFA. That is the standard regex → automaton pipeline.