Why do some planes leave while trails, while others don't?

They're called contrails, short for condensation trails, or vapour trails, and their visibility reveals what conditions are like up at 30,000 feet or so. They're caused by water vapour, ejected by jet engines: as it cools, the water vapour condenses into droplets or ice crystals. That requires just the right humidity and temperature - the drier and colder, the better. So, when you see a high contrail, it means the air up there is dry and below -50C.