In The Sims, and on college lots in The Sims 2: University, they deliver bills every three days. In The Sims 3, and on non-college lots in The Sims 2, they deliver bills on Tuesday and Thursday. They also come by to collect paid bills from the mailbox. In The Sims 2 and The Sims 3, they may also deliver items such as love/hate letters, books and mounted fish.
In The Sims, the mail carrier is known as "Carrie the Mail Carrier". She has dark skin, a wide body, and wears a blue mail carrier uniform. She cannot be interacted with in any way. Like all NPCs in The Sims, she arrives and leaves on foot, and will teleport away if her route is blocked.
In The Sims 2 and The Sims 3, mail carriers become full-fledged Sims, and each neighborhood will have two or three. Sims can talk to them, ask them to hang out, and give them tips. In The Sims 2: Pets, dogs will go to the mailbox to greet them. The uniform they wear is based on that of USPS.