There is more than one factor that caused the different parties in the war to join in.
Catelyn Stark takes Tyrion Lannister prisoner
@BCdotWEB's answer does a good job of summarising this point.
- Catelyn arrested Tyrion based on Littlefinger's accusation WRT the dagger the assassin used to attempt to kill Bran.
- That assassin was hired by Joffrey, who did so to please his father, sort of: to mercy kill the crippled Bran.
- Bran was crippled due to his fall, but he was actually pushed by Jaime because Bran caught Jaime and Cersei in the act.
- The Lannisters are at Winterfell because Jon Arryn got murdered.
- Which was the work of... Petyr Baelish.
However, this only causes the Lannister's to mobilise their forces to try and get Tyrion back no other party actually starts yet.
Eddard Stark learns the truth
This is more the main reason that the War of the Five Kings starts. Ned learns the truth that the three children of Robert Baratheon and Cersei Lannister are actually the result of incest between Cersei and Jaime Lannister. When he finds out about this he intends to make Stannis Baratheon king as he is actually the rightful heir. This is what causes Stannis and Renly Baratheon to join the war as both of them want the throne.
Eddard makes a mistake
As we all know Ned is an "honourable fool" and as such when he learns the truth he confronts Cersei about it. This isn't just beyond stupid but he essentially tells her he plans to get rid of them off the throne and gives away his plans and intentions. This leads to Ned eventually being taken prisoner. This is what causes Robb Stark to join the war, to get his father back.