The most common claim form is the CMS-1500 form. This is what medical clinics use for professional services. (A hospital bills on a separate form, the UB-04). If you work out a Single Case Agreement (SCA) with and insurance company you will fill out the CMS-1500 form.
Below are files with CMS-1500 claim form instructions. Note that most all private insurors follow Medicare guidelines.
The document above is my version of instructions specifically for liposuction for lipedema reimbursement.