The answer is that it depends on your bank and your creditor (whomever you are paying).
Most banks offer "overdraft protection". For a fee, they will process the payment as if you had the money, and your account will become "overdrawn". The next deposit you put in will be applied to the overdraft and once the account is "zeroed" funds will begin to become available. The fees charged, and whether a particular charge will go through, depend on the bank; a few banks will process a payment without a fee if it only happens occasionally, some will hit you with a charge per debit in excess of the amount, while others will charge you per day on which debit transactions were made while your account was overdrawn, and still others will tack on a charge for leaving your account overdrawn for an extended period. In almost all cases, the for-profit institutions (banks) will charge more than non-profit institutions (credit unions).
Watch out for banks that offer "automatic overdraft transfer"; they'll automatically transfer from another account (including, eek, a credit card) to cover a debit to an account with insufficient funds. Sounds great, but the fee they charge is often close to the overdraft fee, while without overdraft protection, if the payment were simply denied at point-of-sale, most banks have smartphone apps that allow for instantaneous transfer of funds, for free.
I've never heard of a "partial payment"; you either have sufficient funds to cover the debit or you don't. If you don't, and you don't have overdraft protection, the bank will simply decline to pay. The result of that situation depends on the exact circumstances; if a utility check bounces, they may try to run it again the next day, or they may immediately cut your utilities and come after you for the balance. At point-of-sale, they simply won't give you the merchandise. Some places are really rabid about pursuing "hot check writers"; if the check doesn't clear you have to pay in cash, on the spot, or they call the police.
As far as "interest on debt", it doesn't work that way; it's a "service fee", otherwise it would be subject to some pretty hefty federal regulation. However, if you figure it as interest, you'd be amazed; let's say you bought lunch at a burger joint for $10. You didn't have enough in the account, but it was close, so the bank let it go - and charged you $35. You have effectively inflated the cost of that meal to $45, a 350% instantaneous price increase (or if you prefer, 350% interest on that $10); I dunno about you but if I were paying $45 for one lunch it wouldn't be at a burger joint.
Most banks simply will not process a check on an already-overdrawn account, overdraft service or otherwise; they'll hit you for the overdraft fee (usually actually a fee for an NSF check) AND decline the payment. So, if you had $1 in your account and wrote a rent check for $1000, it'd still bounce even with overdraft service.