On pretty much every airline I've been on when this happened, the airline just puts you on the next flight. If the next flight is the next day, you are on your own to get a hotel room. If you're lucky, you might get a distressed traveller rate on your hotel room, which is not that big of a discount.
When I was headed to Biloxi on NW for a casino vacation, bad weather caused me to miss the connection and I ended up stranded at MEM. So I actually just gave up and vacationed in Tunica, Miss, instead of Biloxi.
Once when I was stranded at LGA due to rainstorms, I had to go all the way out to the eastern Nassau county to get a room, because everything around the LGA was booked solid that night.