If you've ever taken a ride in an open cockpit aeroplane, I can guarantee that you'll have noticed that the air temperature at altitude is usually not the same as that at ground level, and no doubt realised as well why the old barn stormers wore their leather caps and scarves!
The temperature of the lower atmosphere varies dramatically. In the regions soaring pilots usually frequent, the overall trend is cooling with altitude, but there can be many times when parts of the atmosphere don't cool much, or even increase in temperature as you ascend.

Lapse Rates

The rate of temperature change with height is called the Lapse Rate, and when you measure it in real life, what you're measuring is the "Environmental Lapse Rate" or ELR for short. If you were to  fill a slack, sealed balloon full of air and take it along with you on this climb,  you'd notice two things.  First, the balloon would expand as it got higher, and secondly, if you put a thermometer in the balloon, you'd notice that the temperature of the air would drop as you climbed as well, at a rate of about 3 degrees Celsius (5.4 deg F) per 1000 ft.

This is the rate at which air cools as it rises, as long as it doesn't mix with the surrounding atmosphere and providing that any water vapour within it doesn't condense.  We call this the "Dry Adiabatic Lapse Rate" (DALR).

If the balloon got high enough and cold enough, the water vapour within it would start to condense, and the air in the balloon would cool more slowly (this is because the process of condensation actually releases heat). The rate it will then cool at is about 1.5 degrees Celsius per 1000 ft. We call this the "Saturated Adiabatic Lapse Rate" (SALR).

We've discussed a balloon containing this air, but the same principle holds for a "lump" or "parcel" of air of about the same temperature and pressure not contained in a balloon.   While such a "parcel" of air will be mixing with other air at the edges, we can pretend for many purposes that it behaves a lot like it was in this loose, light balloon.

Stability

In general, a parcel of air will tend to rise if it's at a higher temperature than the surrounding air.

If the Environmental Lapse Rate (ELR) is less than  the DALR, then if you were to take a parcel of air at any altitude and lift it a bit, it would be at a lower temperature than the surrounding air, because it cools at a faster rate, so it would tend to sink back to the surface.  This is a characteristic of stable conditions

If the Environmental Lapse Rate (ELR) is the same as the DALR, then if you were to take a parcel of air at any altitude and lift it a bit, it would still be at the same temperature as the surrounding air, because it cools at the same rate.  In these conditions, if you were to heat a parcel of air a bit, and let it go, it would still cool as it rose, but would maintain a temperature difference, so would continue rising steadily. We would describe the conditions as unstable.

If the Environmental Lapse Rate (ELR) is greater than  the DALR, then if you were to take a parcel of air a and lift it a bit, it would be at a higher temperature than the surrounding air, because it cools at a lower rate, so it would tend to rise very rapidly!  We would rate this atmosphere very unstable indeed, and a layer like this is called superadiabatic.  The very lowest part of the atmosphere at ground level is often superadiabatic.

Insolation

The engine that heats the lower atmosphere is the Sun.  Mostly it does not heat air directly, but passes directly though it (unless it's hazy or polluted), and heats the ground.  The ground in turn, heats the air that touches it by conduction.  Therefore, the atmosphere heats on a daily basis (and cools at night) from the bottom up.  We call the heat available from the sun "insolation".

The air that is warmed by the ground will tend to rise and/or be mixed with other air in the lower levels, and as a result, the ELR at lower levels will become more and more close to the DALR as the day goes on.

We measure the insolation available from the Sun on a daily basis in units of  "Thousands of Degree Feet".

Lapse rate diagrams/traces

Often when soaring, we take send up a power plane in the morning to measure the ELR.  The temperature is measured at 200-500ft intervals, and graphed as shown below. (you can download a PDF file with a preformatted grid for drawing your own temperature traces here)

Inversions

Under some circumstances, the ELR may often be very small, or even negative, meaning that the air is actually staying at the same temperature or warming with height. We call these layers in the atmosphere "inversions". The two most common types of inversion are "Radiation Inversions" and "Subsidence Inversions". (An area where the temperature stays constant with height is called an "isothermal layer")

Radiation inversions are caused by the overnight cooling of the air close to the ground.  The ground radiates away the heat it has absorbed during the day, (and this heat is not transferred to the air). The air in contact with this cooled ground then chills.  In calm, cloudless conditions, the resulting inversion can be extreme.  You may take an early morning power plane flight from a frosty airfield to find that it's shirtsleeves weather only a few hundred feet higher.  This shallow (several hundred feet) extent is typical of radiation inversions.  They are not as marked when cloud is present overnight, as the cloud reflects the radiated heat back at the ground and it doesn't cool as quickly.

Subsidence inversions are are caused by heating of the atmosphere caused by the subsidence of air in a High pressure system, which is commonly where much soaring is done.  The inversion will usually be found at level from about 1000 ft upwards, and will generally lift higher through the day, or even disappear once the lower atmospheric heating overpowers it.  Subsidence inversions can often be seen when soaring when you get near the top, because haze and dust becomes trapped below it. There will be a distinct dividing line between the brown air below and the clear blue above.
 

Cloud Formation

All air contains some water vapour, and the colder the air the less water vapour it can hold.  The ratio between the amount of water vapour air will hold at a particular temperature and the actual amount is called the "relative humidity".  If you cool air enough, even if it has only a small amount of water vapour, eventually that will be all it can hold (100% humidity). Any further cooling, and some of the water will condense into tiny droplets. This critical temperature is called the "dew point".

As a thermal rises, if it can rise high enough, it will cool to the dew point, at which time, some of the water will condense and form cloud. It will continue to rise, but the new lapse rate will be the SALR.
 

Cloud Base

The Cloud base will be where the parcel of air cools enough that it reaches the DEW POINT temperature.  In order to figure that out, we need to know how humid the air is already.  This can be determined by use of a Hygrometer, or by using a dry and wet bulb thermometer.

Thermal Height Prediction from a Lapse Rate diagram

If we can get a good idea of the ELR on a particular morning, and we have a good idea of the likely conditions (clear or overcast), and the expected maximum surface temperature for the day, we can usually make a guess at the likely thermal height. In the simplest example, we'll assume no cloud is expected. We know the predicted surface temperature and we have a good temperature trace.  All we need do is to plot the surface temperature at the bottom of the graph, and draw a line with the same gradient as the DALR.  Assuming a blue day, the thermal height will be the point at which this line cuts the ELR graph! Of course, this is an oversimplification. The example below shows the progression of heating over the day for the typical warming profile, with cloud formation taking place.

Partly adapted from Meteorology for Glider Pilots.