Sometimes planes leave contrails, and sometimes they don’t. It depends on the weather, and specifically it depends on the weather at altitude. It’s also very localized. A plane might leave a trail in one region, and another plane a mile away might not leave a trail.
NASA have put together a contrail forecast page that you can use to roughly predict when contrails are likely for a given region, and a given altitude. The following image should be the latest forecast
The presence of a circle at a particular point simply tells you if the conditions are favorable for contrail formation at the altitude indicated by the color of the circle. The size of the circle only varies so different altitudes can be shown at the same point, but the smaller the circle, the higher the altitude.
The Mb scale on the left is the measure of atmospheric pressure in millibars. This can roughly be translated to altitude, as pressure decreases fairly uniformly with altitude. Planes actually use the air pressure to measure their altitude using an altimeter, but you have to set it to the local sea level pressure in order to get an accurate result for landing and take-off. To avoid confusion, planes flying above 18,000 feet all set their altimeters to the same reference, 29.92 inchs of mercury, or 1013.25mb (for sea level).
The scale starts at 400mb, which is around 23,500 feet, and goes to 125mb, or about 48,500 feet.
The page describes how this works:
Persistent contrails require a relative humidity with respect to ice (RHI) that exceeds 100%. We know that contrails are sometimes observed in areas where estimates of the RHI are less than 100%. The existence of contrails in those locations highlights the “dry-bias” in the humidity fields of the models. Because the input data do not perfectly characterize the meteorological conditions, the diagnoses of persistent contrail conditions are only estimates and will not detect all of the areas where persistent contrails will form and may also add areas of formation that do not exist. All estimates of persistent contrail formation conditions are based on a modified Appleman curve using three different engine propulsion efficiencies. Favorable contrail conditions, for a range of pressure levels between 125 and 400 mb, are represented as concentric circles – color coded with reducing diameter for each level. These results are displayed for engine efficiencies of 0.2, 0.3, and 0.4.
I suggest also looking at a water vapor satellite image.
A more detailed discussion can be found here