Many people do believe that as an employee you need to formally resign from a job by writing a formal resignation letter to your current employer. However, at most companies, there’s no formal requirement that you do so. Most employment in the United States is called “at will”, which means that your current employer can fire you at any time and for any reason (except those that are prohibited under law).
Even those that require you give a certain amount of “notice” before you quit your current job (two-week notice is standard) often may not provide any formal requirement that you do so by writing a letter of resignation or explain in detail why you’re leaving.
So...why might you still want to write a resignation letter?
Many experts believe that writing a formal or even a basic resignation letter when you’re leaving your current job helps make you seem more courteous and professional. However, the real reason to write a resignation letter is to create a paper trail documenting that you have given notice (if your employer requires two weeks notice or some other amount of notice) on a certain date, and to formally kick of the process should there be any legal hiccups (quite rare) with incorrectly processing your final paycheck and last day of benefits.