Death's design refers to a series of events designed by Death itself. Details of Death's designs are revealed by William Bludworth throughout the franchise, but it's not clear how he knows them. During the survival days of Clear Rivers, she was following the design, and marked events related to it on the wall in her room of the Stonybrook Institution Psychiatric Treatment Center.
The people who had the premonition are psychics who were all of the sudden capable of seeing the future.
Order of Death
Those targeted by Death after they survived an event they were originally supposed to die in are killed in a specific order. The order is usually the same one from what should've killed them. The targets are sometimes collectively known as Death's List. Death has killed people through different orders. In Final Destination, the death order was followed the explosion ways in Flight 180. In Final Destination 2, the death order was turned backward of the premonition, which was caused by deaths of the Flight 180 survivors. In Final Destination 3, the death order was still based the order of death, but also on the seating on Devil's Flight and the photos taken gave clues to how they might die. In The Final Destination, the death order was followed the order in the premonition, except Jonathan Groves who died out of order by not changing seats.
Intervening in someone's death can hinder the design, and cheat death, but it will only skip the one who was saved. After the first turn, it will still turn back to the one who was saved before. This rule can't defeat Death at all.
According to William Bludworth, only new life can beat Death. In an alternate ending of Final Destination, Clear Rivers had a baby with Alex Browning (although he is dead), so she and Carter Horton beat Death. In Final Destination 2, Kimberly Corman beat death by attempting to drown in a lake then she "died" for a short period of time before she was resurrected by the doctor thus granting new life and becoming one of the only two survivors who did not die by end of the series.
One of the survivors - Isabella Hudson - had a baby, she was not meant to die on Route 23, so she couldn't help Kimberly and Thomas with "cheating" Death. The birth of a child who was meant to die is likely able to cheat Death, but we have not seen this scenario. Due to the tradition of every single last survivor dying, no matter how hard they try to cheat Death, it is possible that this may never be seen.
Steal another's life
According to William Bludworth, if you kill someone who is not in Death's design, you as the person who was to already to be dead but had cheated Death, receive the remaining life left from the person you killed. Peter eloquently calls this "kill or be killed".
Three instances of this occur in Final Destination 5. Nathan kills Roy, receiving his few remaining days; Roy having an illness that would have killed him shortly anyways. Nathan thus dies shortly after, but not in the original order of death planned by Death, only as the time of death for Roy originally. Peter kills Block, avoiding Death's schedule. Similarly, Sam kills Peter, avoiding Death's schedule. Sam dies shortly after, as Block had been scheduled to die not long after his actual death. There is a bit of a rift here, as Sam was not technically killing someone who wasn't meant to die in the first place.
If it's not your time to die, Death won't let you kill yourself. This happened to Eugene Dix and George Lanter. It was also tested by Carter Horton when he wanted to "gain control" of his life. He drove his car quickly through the traffic trying to kill himself (he would probably kill Clear, Alex and Billy too), but stopped on train tracks waiting for the train to claim his life. Eventually he said "its not my time to die", and unsuccessfully tried to start his car. However, it's revealed in Final Destination 5 that if it's not your time to die, you can't suicide, but the other survivors may be able to kill you. This cannot be proven, since Peter didn't succeed in taking Molly's life, and that it was Peter's turn to die anyway.
In Final Destination 3, Ian McKinley had a theory: if the last person on the list commited suicide, then the rest of the survivors would truly beat Death's design. However, as last one on the list, Wendy Christensen never listened to Ian anyway, it's unknown whether or not Ian's idea is true. However, in an alternate ending of The Final Destination, the last one on the list, Nick O'Bannon thought he could defeat death by commiting suicide.
He jumped out of the mall and landed on a police car and was killed. Nick's action revealed the result of Ian's theory. After Nick's death, Lori Milligan and Janet Cunningham who were the only survivors left on the list, were crushed by an air conditioning unit. This shows that even if the last person on the list kills themselves, the rest of the survivors still can't beat Death. However, seeing as this happened in an alternate ending. It's not meant to be considered canon.