Short answer:
-They strike Seoul with artillery, possibly nuclear weapons. The US loses tens of thousands of people in the blink of an eye, and the South Koreans lose hundreds of thousands. At least. Seoul is virtually destroyed.
-US ground forces do everything they can to hold North Korean advances to a stalemate in the first 30 days. American ground forces hold the line, although it's a really, really shitty month.
-The US launches a massive air and naval campaign. American planes bomb the ever-living shit out the North Koreans, destroying troop and equipment depots, air defense systems, and political refuges. Naval gunfire launches shells the size of VW bugs everywhere that has any value within 30 miles of the shore.
-US ground forces end this fight within the next 30-60 days. After about 4 months of war, the combat is over. The North Koreans have lost over a million people. The South Koreans have lost even more, and US military deaths are close to 15,000.
-North Korean regime change occurs. Kim Jong Un is executed as a war criminal, the government is in total disarray, and China is scared to death. They have more refugees trying to cross into their country than the US has illegal immigrants, and they don't know what to do with all the wounded, starving, and the poor. It takes years and billions of dollars to try and rebuild the country. It happens, but it's slow. And at the end of the day, the US gets a military base on the Chinese border. They hate that, and the US finally has the Chinese by the balls again.