I will try to give an elementary explanation and an elementary example, and will not go into details b/c of the current rate levels etc.
An inverted yield curve is based on the U.S Treasury rates. U.S Treasury rates go from anywhere from 1 day (the shortest) to 30 yr. bonds (the longest). Generally speaking when you loan your money to the U.S. Gov't (which is what you do if you buy a treasury) the longer the term usually gives you a higher rate, which makes sense b/c the longer you tie up your money the better rate you should get. This is basically how old school banks make money. You having a savings account and they pay you an overnight rate. They then take that money and lend it long term, say for a morgtage. But if they have to pay more to you than they can loan it out for, they can't make money. This won;t happen in reality, but it is one of the things that was happening in 2003-2008 and was part of the credit crisis that caused the modern day depression. So when the shortest rate is the lowest and progressively gets higher the longer in time, this is called a normal yield curve. An inverted yield curve is when any of the shorter duration rates is higher than any longer term rate. For example the 3 month T-Bill has had a much higher yield than the 2, 5, and 10 year T-notes for quite a while now, thus the yield curve has been inverted for a long time now. However today for a brief moment, the 10 year T-Note had a rate lower than the 2 year T-note, thus an inversion in the 2-10 spread. The reason it is all over the papers is that this particular inversion is closely followed by many and often seen as a good indicator of a recession to come. I personally think that the 3 month to 10 year is a better tell, but that is another topic. So in short an inverted yield curve is when a shorter term maturity instrument carries a higher interest rate than a longer term maturity instrument in that same family.
Oh and the simple answer as to why the panic is that it has been that recession indicator 6-24 months down the road.