[bisonbison]

Wiki, not surprisingly, is not very clear. From what I can gather:

Truman wanted to run. The main contender was Kefauver. Kefauver won NH and Truman dropped out. Then Barkley started to campaign and went nowhere. Kefauver won 12 of 15 primaries but was blocked by the party heads at the convention who picked Stevenson.

Stevenson of course got destroyed by Ike, as would have happened to anyone. Then 4 years later he got destroyed again, only worse.