The bubble of relatively warm air that rises upward from the surface is called a "thermal".
In hurricanes, condensation occurs through a process known as CISK (Convective Instability of the Second Kind). In CISK, surface convergence causes rising motion around a surface cyclone. The air cools as it rises and condensation occurs, which releases latent heat into the atmosphere. This heating causes air to expand which creates high pressure above. The force resulting from the established pressure gradient causes air to diverge at upper levels. .
Since pressure is a measure of the weight of air above a unit area, removal of air at upper levels subsequently reduces pressure at the surface. A further reduction in surface pressure leads to increasing convergence, which further intensifies the rising motion, latent heat release, and so on. Despite the absence of fronts, a tremendous amount of lifting occurs in hurricanes, with intense condensation leading to the development of deep clouds and heavy rainfall. .
Fronts are boundaries between air masses. Depending on the air masses involved and which way the fronts move, fronts can be either warm, cold, stationary, or occluded. In the case of a cold front, a colder, denser air mass lifts the warm, moist air ahead of it. As the air rises, it cools and its moisture condenses to produce clouds and precipitation. Due to the steep slope of a cold front, vigorous rising motion is often produced, leading to the development of showers and occasionally severe thunderstorms.
In the case of a warm front, the warm, less dense air rises up and over the colder air ahead of the front. Again, the air cools as it rises and its moisture condenses to produce clouds and precipitation. Warm fronts have a gentler slope and generally move more slowly than cold fronts, so the rising motion along warm fronts is much more gradual. Precipitation that develops in advance of a surface warm front is typically steady and more widespread than precipitation associated with a cold front.