Sources of lift
In the atmosphere, rising air creates what we know as "weather". Air rising leads to the formation of clouds and precipitation, be it a dull overcast day of stratus and drizzle in the autumn or winter, a day of sunshine and showers in the spring, or of thunderstorms in the summer. Whatever the case, both the initial stability of the atmosphere and potential lifting mechanisms are important to detemine the extent and type of weather that might occur. 
 
Isentropic ascent. This is arguably the most important concept to grasp when investigating into methods of ascent. An initial understanding of fronts will yield the conecpt that warm air must rise over cold air because it is less dense. However, another and perhaps more appealing method, is to approach this from a thermodynamic standpoint. It turns out that a line of constant potential temperature is a line of constant entropy, and these contours are called "isentropes". The advantage of this method, known as isentropic analysis, is that is provides an explicit representation of vertical motion on horizontal maps. Therefore, we now have a visual repersentation of why warm air must ascend over an advancing cold front - it must conserve potential temperature (and therefore entropy) and rise along isentropic surfaces. This then leads onto the idea that whenever there is warm air advection (WAA), the air must be ascending since the isentropic surfaces are sloped (although to a much lesser degree than with a front). Therefore, air in the warm sector of an approaching depression is generally rising, which can be conducve for shower and thuderstorm development in the summer. Ascent of air over a cold front is generally most rapid, since the isentropic surfaces are sloped most steeply here. The diagram below shows how air will rise over a front - this could be either a cold front, moving from left to right, or a warm front, moving from right to left. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Here, a front is represented by sloped isentropic surfaces. In this case, the parcel of air does not follow the isentropic surface, beacuse it becomes saturated, and the release of latent heat allows its potential temperature to increase. 
From http://derecho.math.uwm.edu/classes/SynII/IsenAnalysis.pdf
The diagram shows that isentropic ascent is only truely followed in a completely dry atmosphere. The parcel reaches saturation at the LCL at point 2, after which it ceases to follow an isentropic surface. However, it is still a relatively good approximation for the Earth. Pressure increases along the y axis and potential temperature increases along the x axis. So, the parcel is moving from high pressure at the surface to lower pressure aloft as it ascends above the front. This is important, because when drawn on an isentropic analysis chart, wind blowing from high pressure levels to low pressure levels indicates rising motion, while wind blowing from low pressure levels to high pressure levels indicates sinking motion.
 
Fronts. Although the underlying theory of why warm air ascends over cold air in a front has been explored, it is still useful to explore the fronts more generally and look at the different weather they produce. 
 
Fronts provide a form of dynamical forcing, by forcing cold air to rise over warm air. However, they tend to be shallow - that is to say they do not have a steep gradient. A typical warm front will have a gradient of 1 in 400, and a typical cold front will have a gradient of 1 in 100 to 1 in 40. Of course, they can be much steeper than this and the rate of ascent also depends on the speed at which air flows over the front as well as the gradient. A cold front is more likely to procuce convective clouds and precipitation, such as showers and thunderstorms, because the steeper frontal surface forces air to rise rapidly. On the other hand, a warm front will produce stratiform rain and cloud that gives the characteristic drizzle and grey cloud day common in the UK during the winter. 
 
Cold fronts can sometimes be associated with a long line of thunderstorms known as a squall line. These generally form when there is instability already present in the atmosphere, with which the cold front acts as a trigger mechanism upon which thunderstorms can form. 
From http://www.physicalgeography.net/fundamentals/7r.html
 
Warm fronts usually produce more benign weather, although the warm airmass introduced behind them can introduce instability, and with warm air that is rising, as mentioned above, thunderstorms can initiate - especially in the summer. 
From http://www.physicalgeography.net/fundamentals/7r.html
Convergence. Convergence of air can occur at any height within the atmosphere, but is most likely near the surface, where winds are least uniform in direction and speed. Thus, Low-level convergence is generally most important. When air collides or converges at the surface, it is forced to rise, since there is nowhere else for it to go. Large-scale convergence can occur between extratropical cyclones and anti-cyclones and can lead to the formation of a large area of cloud and rain. Frictional convergence is probably one of the most significant methods at the surface: when air travels over land directly after blowing off the sea, it must slow down rapidly due to the increase in friction over the land compared to the sea. This causes frictional, or speed convergence and forces air to ascend. Coupled with a sea breeze, this can enhance clouds and potentially even produce precipitation. Alternatively, two sea breezes moving towards one another and colliding can produce spectacular convergence. This is best known in Florida, but can also occur in south west England on a warm summer's day. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Convection. Although not a widespread source of lift which is the case with the other examples, convection can produce extremely heavy precipitation over localized areas. Convection occurs because the atmosphere is a fluid, allowing a spatial distribution of air with different densities and volumes to exist. The least dense air will rise above denser air around it since a lower force is exerted on it due to gravity. Solar insolation heats up air from the surface, creating instability as a result of differential heating of the land. The advection of warm air can also create instability by increasing the temperature of an entire layer of air. Convective instability can be enhanced by any number of lifting mechanims to produce more vigorous showers and thunderstorms. 
 
Vorticity advection. This is a fundamental method by which air can ascend over a large area. A measure of the rotation of a fluid about an axis, vorticity therefore exists along the meandering bends of the upper-level jet stream, known as troughs (associated with low pressure at the surface) and ridges (associated with high pressure at the surface). The advection of vorticity occurs downstream of these bends, where there is a gradient of vorticity to be moved with the flow. Positive vorticity is taken to be anti-clockwise, or cyclonic, by definition - and so this is advected downstream of a trough. Negative vorticity is therefore advected downstream of a ridge, and these are known as cyclonic vorticity advection (CVA) and anticyclonic vorticity advection (AVA) respectively. (Or positive vorticity advection and negative vorticity advection.) Without introducing too many more complicated terms, CVA produces divergence of the air aloft, which in turn produces convergence at the surface and ascent throughout the atmosphere. In contrast, AVA produces convergence of the air aloft, and so divergence at the surface and descending air in-between. 
 
The Intertropical Convergence Zone (ITCZ) is probably the best known convergence zone in the world. Here, the trade winds converge near the equator to produce towering cumulonimbus clouds that can be seen in this satellite image. 
From http://earthobservatory.nasa.gov/IOTD/view.php?id=703
A diagram depicting the polar jet stream in the Northern hemisphere over Europe and the Atlantic Ocean. "PVA" and "NVA" repesent positve vorticity advection and negative vorticity advection, respectively. At the surface, below the PVA aloft, a low pressure area is likely to form. Below the NVA aloft, a high pressure area is likely to form at the surface. 
From http://www.meted.ucar.edu/EUMETSAT/vorticity/navmenu.php?tab=1&page=1.1.0 
Jet streaks. A jet streak is a region of increased wind speed within the jet stream. They only esist for part of the distance of the jet stream, and so they have entrance and exit regions - where the air flows into them (and increases speed) and flows out of them (and decreases speed). When the jet stream moves in a straight line, divergence of air aloft occurs at the right entrance region and left exit region. Convergence of air aloft occurs under the left entrance region and right exit region. Therefore, a low pressure system at the surface is more likely to develop under the right entrance and left exit regions, while an area of high pressure at the surface is more likely to develop under the left entrance and right exit regions, at the surface. When a jet streak becomes curved, vorticity advection then comes into play, with an enhancement of convergence and divergence aloft occuring where the effect of the jet streak and vorticity advection constructively intefere with one another. The main principles of jet streaks will be left for a discussion in another section, although some diagrams displaying where upper-level convergence and divergence will occur are displayed below:
These diagrams which show where convergence and divergence occurs in the upper levels (above 500 hPa or 6000m, typically). The three main types of jet streak are displayed, although in both the cyclonically and anticyclonically curved cases, the divergence and convergence is enhanced compared to the straight-line case. On the other side of the jet streak, there is little acceleration of the air, because the effect of the jet streak and vorticity advection cancel each other out. 
From https://www.meted.ucar.edu/search/module_images.php?currentPage=showAll&module_id=166