What are fixed income derivatives? And how were they one of the reasons for the 2008–2009 financial crisis?

A fixed income derivative is a type of financial instrument with a value that is derived (hence the term) from some sort of fundamental underlying asset. Prominent examples of this include credit default swaps, interest rate swaps, futures contracts, and forward rate agreements.
A credit default swap is akin to an insurance agreement where the seller promises to compensate the buyer if a borrower mentioned in the agreement defaults on their obligation.
An interest rate swap involves an agreement where two (or more) parties exchange one stream of future interest payments for another based on some predetermined amount of principal.
Futures contracts involve the promise to deliver some commodity at a given price on a predetermined settlement date sometime in the future. Such contracts are traded on futures exchanges and can be closed out before the settlement date.
Forward rate agreements allow two (or more) parties to specify the rate of interest to be paid at some settlement date in the future. They are cash-settled and determined based on the difference between the rate referenced in the agreement and the market reference rate.
Fixed income derivatives were one contributing factor to the 2008–2009 financial crisis because banks were using them heavily in hedge fund trading as the financial industry gradually deregulated. Mortgage-backed securities were the derivative of choice for many, with banks essentially lending to each other through the sale of derivatives.
These seemed viable because, according to Kimberly Amadeo on The Balance,

the contract's seller doesn't have to own the underlying asset. He can fulfill the contract by giving the buyer enough money to buy the asset at the prevailing price. He can also give the buyer another derivative contract that offsets the value of the first. This makes derivatives much easier to trade than the asset itself.

However, in order to prop up the sale of so many derivatives, the banks needed to sell more mortgages. Qualified borrowers already had mortgages, so the only place the banks could turn was to subprime borrowers who weren't really qualified to have a mortgage in the first place. The housing market, as a result, temporarily exploded.
Eventually, however, housing prices started to fall as supply eclipsed demand, creating a volatile situation where subprime borrowers who were financially underwater couldn't afford to pay their mortgages or sell their houses. The rash of defaults eroded the value of the derivatives, and spooked banks stopped lending to one another, ultimately resulting in the crash.
https://www.thebalance.com/what-are-derivatives-3305833