Term
Server
The player who hits the serve.
The player who hits the serve to start the rally (Section 2, Definitions). In doubles, the two players on a team usually take turns being the server (5.B, first server, second server).
After scoring a point, the server moves to the other serving position (left or right) and serves again (5.A.2, 5.B.5, After scoring, only the serving team switches sides).
See it in a figure
This figure is from the rule "Serving order in doubles".
How to read the sources: numbers like (8.F.4) are rulebook sections, and links are outside sources. An "Author's note" is the author's personal experience, not an official rule.