If you buy $2000 worth on your credit card and don't pay that $2000 off by the due date, then you will be charged interest on the entire balance - even if you made a partial repayment of $1999.
If you pay the bill late, then most credit card companies will charge interest back to the purchase date (back dating interest).
If your bill is overdue, then most will cancel your interest free days until the overdue balance is paid completely.
Some credit card companies will charge interest on the entire balance even if you've paid a portion of your bill (even the new transactions that hasn't been billed to the statement yet). So until you pay your credit card statement in it's entirety, you will be charged interest even on the portion that you have partly paid off. And lose the interest free days on all the new transactions until the entire statement balance is paid off. This is entirely unfair but this is how they operate.
Balance transfer deals involves transferring your debt from one credit card provider to a new one who might be offering six months interest free deal. When the special period ends (in this example, it's six months) and there is a transfer balance remaining, the interest will be charged on that balance as if it was a cash advance. Cash advance rates are usually much higher than transaction rates.
If you find that you have overlooked the bill and paid it 1-2 days late (or even up to 1 week late) by accident and you have a good payment history, phone your card provider to explain and request them (very politely of course) to reverse the late fees and interest charges as a courtesy to you. They will usually reverse it for you if you have a good reason or you have a good payment history. It is usually left to the discretion of the staff member that is working which is why it pays to be polite when calling them to reverse any charges and fees.
If that doesn't motivate you to pay off the credit card statement in full, then you shouldn't be using a credit card. Not when it's costing you 10-28% extra in terms of interest charges.