Character Analysis
The Phantom Menace marks the fourth appearance of R2-D2 and C-3P0, the only characters to have appeared in every Star Wars film to date. (And yes, we're counting Star Wars: The Clone Wars 2008 theatrical release.)
Since this film is the first chronologically, R2-D2 and C-3P0's appearances serve as origin stories for the characters and to set up their place in the prequel trilogy.
When we first meet R2, he's serving the Queen aboard her Nubian Cruiser and saves the ship by reconnecting the shield generator. For the remainder of the film, the droid serves Amidala. He even joins Anakin in the Naboo Fighter to assist the young pilot in the battle against the droid control station, paralleling his role helping Luke during the Death Star assault years later.
In The Phantom Menace, we learn that Anakin Skywalker built C-3P0 to help his mom around the house. Yep, Darth Vader built a robot for his mommy as a present. Kind of takes the menace out of the character, doesn't it?
This also means Anakin is responsible for C-3P0's Jeeves-like personality. Either way, C-3P0 assists Anakin in preparing the pod for the race and remains on Tatooine with Shmi Skywalker.
Although neither character serves much purpose in the film, they do have a few moments where they slip into their former roles as the film's comic relief:
C-3P0: Oh, hello. I don't believe we have been introduced.
R2-D2: [Excited Beeping.]
C-3P0: R2-D2. A pleasure to meet you. I am C-3P0, human-cyborg relations.
R2-D2: [Chirping, Beeping]
C-3P0: I beg your pardon, but what do you mean, "naked"?
R2-D2: [Chirping]
C-3P0: My parts are showing? My goodness! Oh!
Although it's technically their first meeting, it's a familiar formula to Star Wars fans. In double act fashion, R2-D2 sets up the joke with his chirps and beeps, and C-3P0 plays the straight man while delivering the punch line. Classic stuff.