It's easier to define what they are, then to determine what they really mean.
The deficit, is the amount of money the federal government spends over and above what it takes in in revenue. It's the rate at which we are adding to the federal debt.
The federal debt is the cumulative amount that the government owes total.
There are a few reasons why our government tends to run a deficit. One is that politically, it's easy to spend money and hard to take it in. Everyone (or at least most people), like government services, but few like taxes. So we tend to spend more than we take in.
From an economic standpoint, a deficit also boosts the economy (at least temporarily), by pumping money into the economy. The opposite situation (where taxes exceed spending - a surplus) takes money out of the economy. So it tends to boost economic growth. Some economists believe in "countercyclical deficits" - that is, we should deficit spend during recessions, to boost the economy, then balance the budget (or generate a surplus), when the economy is good. But usually, we are able to boost deficit spending during down times, but lack the political fortitude to cut back on them during good times.
What deficits mean in real terms is actually a very good question. Historically, we've never actually paid down our federal debt at all - we've just "outgrown it". By that I mean that our economy has grown in size to keep our debt at a somewhat stable level in terms of GDP, despite consistently running deficits.
The US has a few things going for it - one is that there is a robust demand for US Government debt. The deficit is financed by the sale of US Treasuries, and despite our growing indebtedness, there's no sign of demand for Treasuries slowing. It's still the safest place to park your money. Also our debt is in our own currency, so we don't have to worry about exchange rate risk.
One of the greatest possible risks with high debt is inflation. But that's relatively under control too. So, it's a real question - do deficits matter? While this is a valid economic question, it's also really politically loaded.
Historically, Republicans have been 'deficit hawks' - fiercely critical of deficit spending, but lately that seems to be the case only under a Democratic president.
So how much deficits really matter is an open question, but certainly, the increased deficit spending we are doing currently raises the stakes as to the answer of that question.