If the stock market crashes 20%, do bonds suffer or is there little to no correlation between the two markets?
It depends. Very generally when yields go up stocks go down and when yields go down stocks go up (as has been happening lately).
If we look at the yield of the 10 year bond it reflects future expectations for interest rates. If the rate today is very low but expectations are that the short term rates will go up that would be reflected in a higher yield simply because no one would buy the longer term bond if they could simply wait out and get a better return on shoter term investments. If expectations are that the rate is going down you get what's called an inverted yield curve. The inverted yield curve is usually a sign of economic trouble ahead. Yields are also influenced by inflation expectations as @rhaskett is alluding in his answer.
So. If the stock market crashes because the economy is doing poorly and if interest rates are relatively high then people would expect the rates to go down and therefore bonds will go up! However, if there's rampant inflation and the rates are going up we can expect stocks and bonds to move in opposite directions. Another interpretation of that is that one would expect stock prices to track inflation pretty well because company revenue is going to go up with inflation. If we're just talking about a bump in the road correction in a healthy economy I wouldn't expect that to have much of an immediate effect though bonds might go down a little bit in the short term but possibly even more in the long term as interest rates eventually head higher. Another scenario is a very low interest rate environment (as today) with a stock market crash and not a lot of room for yields to go further down.
Both stocks and bonds are influenced by current interest rates, interest rate expectations, current inflation, inflation expectations and stock price expectation. Add noise and stir.
The short answer is if you own a representative index of global bonds (say AGG) and global stocks (say ACWI) the bonds will generally only suffer minimally in even the medium large market crashes you describe. However, there are some caveats.
Not all bonds will tend to react the same way. Bonds that are considered higher-yield (say BBB rated and below) tend to drop significantly in stock market crashes though not as much as stock markets themselves. Emerging market bonds can drop even more as weaker foreign currencies can drop in global crashes as well.
Also, if a local market crash is caused by rampant inflation as in the US during the 70s-80s, bonds can crash at the same time as markets. There hasn't been a global crash caused by inflation after countries left the gold standard, but that doesn't mean it can't happen.
Still, I don't mean to scare you away from adding bond exposure to a stock portfolio as bonds tend to have low correlations with stocks and significant returns. Just be aware that these correlations can change over time (sometimes quickly) and depend on which stocks/bonds you invest in.