Is "singing telegram" a trademarked phrase? Or did they just think players are too young to know what a "telegram" was?
At any rate, "Singer" is a Profession, which means that you pursue it on your own time (much like Architect or Ghost Hunter). You advance in the profession by playing gigs, singing for tips in public, and delivering "Sing-A-Grams." (Is "singing telegram" a trademarked phrase? Or did they just think players are too young to know what a "telegram" was?)
Singing for tips is a straightforward process. If you put a guitar in your Sim's inventory, they have the option to strum it to accompany their singing. Just pick a spot in public, click your Sim, and start wailing away!
Tip: it's worth your time to spend a few minutes scanning around town until you find a group of townies, then send your Sim to their location. I have had many fruitless sessions where my Sim sang for hours and hours and no one stopped by to listen to her performance!
This profession feels very similar to Ghost Hunter. You get messages that someone needs a Sing-A-Gram, and a map tag showing you their location. Proceed to that location and deliver the Sing-A-Gram, which is either a "cheer up" or a "congratulations," depending on the occasion. This is also a great way to meet other Sims, since it gives you instant access to townies when they are not otherwise occupied.
To play a gig, find a Proprietor and audition for the part. If the Proprietor accepts your Sim's audition, they will be booked for a gig.
Your Sim can earn chunks of money with all three of these activities, but they also get a weekly stipend. In theory, this profession would be a great way for a broke Sim to make a lot of money fast without requiring too much talent or player investment.
On the flip side, there aren't any skills that your Sim can boost in order to improve their job performance. You also can't become a singer until you turn Young Adult. All of which means that if you have a baby that you think is destined to be a pop star, there is no prep work you can do while they are children.