Steps of a Query (simplified)

Client (aka application) connects to the backend (aka database server) [cached connections]
Client transmits a SQL statement to the backend
Parser attempts to read the statement. If errors are found, it stops and bounces them to the client
If no errors were found, a parse tree is created. This is an internal structure that breaks apart your query into a logical, simplified form. Similar to the "adjective - noun - verb" charts from school. Actually looks like a tree or a nested set
The rewrite system changes the parse tree as needed if there are any rules or views involved in the query
Now that the system knows exactly what to fetch, it needs to figure out how to get that information back to the client, in as timely a manner as possible. This is the job of the planner/optimizer
The planner tries out every possible way of "running through" the parse tree. Some simple queries only have one possible path through the parse tree, but most will have many. Each plan is assigned a cost, and is the job of the optimizer to pick the best one. [scanning and joining]
TSP and GEQO
See src/backend/optimizer/geqo/geqo_main.c (a "Genetic Algorithm query optimization")
The winning plan is handed to the executor, which recursively descends the tree, then returns the information from the top node of the tree to the client/application