Tornadoes are perhaps the most well-known of all severe weather phenomenon. Almost everyone has head of a tornado and the destructive winds that they produce. In general, they form at the base of a cumulonimbus cloud (or thunderstorm), in the updraft region, and extend from the lowered cloud base to the surface. They can be observed as a rapidly rotating cloud (known as the funnel), or from rotating dust and debris close to the surface. However, by their nature, tornadoes are unpredictable and difficult to study. As such, there are many theories regarding tornadogenesis, but there is no concrete method known that tornadoes form by - in this section I will explain some of the prominent theories.
From a popular meteorology textbook by Markowski and Richardson: "Tornadoes are violently rotating columns of air, usually associated with a swirling cloud of debris or dust near the ground and a funnel-shaped cloud extending downward from the base of the parent cumulonimbus updraft." This is almost true, except that not all tornadoes are associated with a funnel cloud.
A classic, cone tornado in the Great Plains of the USA.
The fundamental property of a tornado is that it occurs in association with a thunderstorm. Becuase tornadoes are rotating columns of air that extend verticically into the parent thundertorm, they require the updraft of the thunderstorm itself to be rotating - a                   thunderstorm. Tornadogenesis requires that a large vertical vorticity arises at the surface. The tilting of horizontal vorticity (due to speed shear close to the surface) into vertical vorticity, rotating in the horizontal plane, due to the presence of an overhead thunderstorm updraft, is not effective at produing vertical vorticity near the surface. This is the process that forms the mid-level mesocyclone in a supercell thunderstorm - see         section.
It is important to realise that a tornado is essentially a rapidly rotating updraft. The rate of spin can be increased if this column is stretched, thinning the column, but making it rotate more rapidly, and increasing the vorticity of the column. This is known as the conservation of angular momentum and is best visualised by the classic example of an ice skater drawing their arms in to rotate more rapidly. 
First, let's look at where we would expect a tornado to form with respect to the parent supercell:
Above is a schematic and a simple radar depiction of a classic supercell. 
In the radar echo, the "hook" at the bottom of the storm is caused by rain and falling precipitation getting wrapped around the mesocyclone and falling at the rear of the storm. This is known as the rear flank downdraft.
From Markowski and Richardson textbook: Mesoscale Meteorology in Midlatitudes, 2010)
The tornado generally forms at the rear of a passing supercell (ie, the last part to move overhead). In the example given above, the storm is likely to be travelling north-east. The tornado forms close to the boundary between the warm updraft air and cold rear-flank downdraft air behind. In fact, there is significant evidence to suggest that the downdraft plays a pivotal role in tornadogenesis. It is generally regions at the surface that miss the worst of the precipitation (including massive hail) that are impacted by the trailing tornado. 
Vortex line demonstration of why a downfraft is needed for the production of significant vertical vorticity close to the surface. Image (iii) indicates the tilting of vorticity by the updraft, which generates the mid-level mesocyclone. 
The diagram above show how important the downdraft of a thunderstorm is thought to be in tornadogenesis. Essentially, the downdraft moves (or advects) the vertical vorticity, created by the updraft tilting, closer to the surface. Compression of the vortex lines may also occur as a result of the downdaft influence, which can aid the formation of a tornado. Once a tornado has formed, tilting of the preexisting horizontal vorticity by the tornado updraft itsself probably contributes significantly to the near-ground vertical vorticity. If the downdraft brings the vertical vorticity very close to the surface then the rapid acceleration of air away from the ground (where the vertical velocity must be zero), can stretch the vorticity column, so increasing rotation further, helping to reinforce a forming tornado. This may be the reason that the strongest wind speeds in tornadoes tend to be close to the surface. 
Vortex line demonstration of how a tornado can arise from convergence alone, in the absence of a downdraft. This assumes there is preexisting vertical vorticity present at the ground. 
In the above example, a tornado is theoretically able to form due to converging winds compressing vortex lines. This is equivalent to stretching the column of air, and due to the conservation of angular momentum, results in an increase in the spin rate and possibly the formation of a tornado. Interestingly, at the cloud base, the sudden release of latent heat from water condensing into cloud droplets rapidly increases the buoyancy of the updraft air, and thus the vertical velocity. This results in a stretching of the vorticity column at the cloud base, which increases the vorticity and spin rate and can enhance tornadogenesis from aloft. 
A classic wall cloud hanging from the base of a supercell mesocyclone.
The wall cloud base can be only a few hundred metres off the ground. 
A wall cloud (above) is thought to form due to updraft air entering the storm after passing through the rainy downfraft. This moistens the air and so lowers the pressure level at which it condenses, forming a lower hanging area of cloud in the main updraft. Another theory is that the low pressure in the centre of the updraft, generated by the rapidly rising air, allows the air to condense at a lower height (and lower pressure, relative to the surrounding air). In any case, the low hanging wall cloud encourages the stretching of the updraft column (as a reult of increased buoyancy from latent heat release), to occur closer to the ground, thereby increasing the vorticity at lower levels. This will encourage tornado develpment and help sustain a tornado for longer. 
Idealized evolution of vortex rings and arches superimposed on a photograph of a supercell thunderstorm. This illustrates another important role the downdraft could play in tornadogenesis. 
The votex lines at 1), 2) and 3) indicate horizontal vorticity that has been generated at the edge of the descending downdraft air. This is known as a baroclinic generation of vorticity, as it is generated at the boundary between warm and cold air. The presence of negative buoyancy causes the vortex rings to sink towards the ground as they are generated. These may then be swept forward into an updraft in close proximity, which can tilt the vortex rings and stretch them upward. This leads to arched vortex lines and a couplet of cyclonic (C) and anticyclonic (A) vertical vorticity columns: 
Schematic of the upward tilting of horizontal vorticity (by the updraft), generated by the sinking downdraft air. 
The problem is that these vorticity columns produce two rotating updrafts that feed into one storm: which way does the storm spin? Fortunately, through some complex mathematics, it is found that under normal wind shear conditions for supercellular development (ie, wind veers with height in the northern hemisphere), then cell growth is enhanced on the right flank of the storm, and this is where the updraft is strongest. This is why supercells are often called "right movers" because they move to the right reletive to the background flow, as a result of this enhancement on the right flank. Therefore, in the northern hemisphere, supercells and tornadoes tend to rotate cyclonically - the opposite is true for the southern hemisphere. Anticyclonically rotating storms are possible in the northern hemisphere, but they tend to be weaker.