Conventional convertible bonds give the bondholder the right to exchange the bond for a certain amount of the issuer's common shares, regardless of the current market price of those shares. The bondholder will see a large payoff in situations where the total value of the underlying common shares appreciate far beyond the value of the bond. For example, ABC Corp.'s convertible bond is worth $1,000 and can be converted into 20 common shares. A bondholder would convert the bond when the share price exceeds $50 ($1,000/20). In some ways, convertible bonds have a payoff that is similar to that of holding a call option.
Death spiral convertible bonds differ from normal convertible bonds in the sense that instead of having a predetermined, fixed conversion ratio, death spirals have a floating conversion ratio in which the holder receives a discount for converting shares. For example, a death spiral bond with a face value of $1,000 has a convertible value of $1,500, which means that a bondholder will receive $1,500 dollars worth of equity for giving up a $1,000 bond.
The problem with this class of convertible security is that these bonds promote large drops in share price. This is because once a bondholder converts the bond into shares, the influx of these new shares dilutes the share price. Furthermore, bondholders have a large incentive to short sell the common stock prior to conversion, knowing that the share price will fall after the conversion is made. The short sale causes another drop in share price, giving this security its 'death spiral' name.
Interestingly enough, death spiral convertibles were a common form of financing for companies that were badly in need of cash and had little means of raising it. These companies felt that the prospect of borrowing money without paying it back outweighed the losses incurred by decreased share values.
For further reading, check out Convertible Bonds: An Introduction.