A blood bowl is a device used to reduce blood transfer in tandem suspensions. When one person is suspended from another, blood can run down the suspension ropes and into the hook at the other end, thus infecting the second person with anything in the first person's blood. Because of the amount of blood involved and the ease of entry, transmission rates for diseases like hepatitis are very high.
A blood bowl attempts to solve this problem by "catching" the blood and not allowing it to continue down the rope. In general, it is considered extremely irresponsible to perform a tandem suspension without these unless the people are consenting fluid partners. It should also be noted that even when precautions like these are taken, even in the case of the most static of suspensions, the risk of blood transfer between suspendees can not be entirely eliminated.
But you can minimize that risk, and you should.