The Common App's dominance over the last 25 years means students are applying to more colleges and the chance a waitlist spot comes your way increases.
Do NOT treat the Wait List as a rejection, even at the most competitive colleges. Yes, some institutions will barely visit their waitlists, others may depend on the darn thing. You don't know what you don't know.
Read the waitlist instructions carefully. If they open the door to reccomendation(s), try to secure another from someone who did not submit anything on your behalf when you first applied -- consider the same pool that's widely suggested for the initial application round: a supervisor at work, a coach, a teacher. Some colleges welcome a letter or essay to confirm interest in their school. Unless the institution is completely off the table for you, do try to reach out to them in the manner suggested.
The essay or letter to admissions should express gratitude for keeping you under consideration. Absolutely reiterate your main reasons for applying as a reminder of who you are and how you have defined yourself as a candidate (rephrased from what appeared on your application).
And critically, supply NEW information if you can; achievements that took place after you submitted the application. Are you the star of the spring musical? Did you earn your Eagle Scout badge? Were you appointed class speaker for graduation?
Do not beg! As with the personal statement essay, this is a dialogue between you and admissions. It's like you're trying to convince said college that you'd be excellent to live with, a stellar roommate. You're engaged in life, you've got interesting stuff going on, you care and you follow through.