If someone pays me with a Credit Card or Check, but they don't use my website to make the payment. Who do they sign the check to, or make the Credit Card payment to?

If you are selling directly to the end user they can pay you with a check or money order. They would make the check out to your business name. If you do not have a business name they need to make the check out to your personally so you can deposit it.

If you are selling offline and they want to pay with a credit card then you will need to have a merchant account or you can pull out your phone visit your website and go through the process like an online order.