Debt is accumulated deficits. You can't pay down the debt if you have deficits. Since bush (who was given a surplus when he took office) ran the debt so high then crashed the economy, it takes time to reign in deficits so you can pay down debt.
If the government spends more than it takes in over the course of one year, then it has run a deficit. A deficit applies to just one year.
So, if the government takes in $10 trillion dollars but spends $13 trillion dollars in one year, then it has run a $3 trillion dollar deficit.
When the government runs a deficit, then it must borrow money to make up the difference.
A debt is completely different. Think of debt as accumulated deficits.
If the government has to borrow money every year, then its debt will continue to grow year-after-year. This debt does not disappear unless the government elects to try and pay it down (rare occurrence)
The debt usually grows year-after-year. With each additional deficit, the debt continues to grow.