I am confused about this thing. In the temperature table, the first
column explains specific temperature and the second column explains
corresponding saturation pressure.
Correct. Here you are referring to the Saturated Water-Temperature Table where the first column is the saturation temperature and the second column is the corresponding saturation pressure.
From what I understand, if we use T-v table, the property that is kept
constant is pressure.
You aren't thinking about this correctly. When dealing with saturated water, for every saturation temperature there is a unique corresponding saturation pressure, and vice-versa. In other words, when dealing with saturated water, temperature and pressure are not independent variables.
Then, at the given constant pressure, the saturation temperature is
the temperature when a substance changes phase (for example from water
starts to boil at saturation temperature 100°C in constant pressure of
True, but you can just as well say that, at a given constant temperature, the saturation pressure is the pressure when a substance changes phase. For example, you can say that water starts to boil at a saturation pressure of 1 atm at constant temperature of 100 deg C. It means the same thing.
Then why in some textbooks, they use T-v diagram for temperature table
instead of P-v diagram? isn't it the temperature that is kept constant
in this table?
You are confusing the tables with the diagrams. There are two regions in the T-v and P-v diagrams, one region that covers the saturated water table (discussed above) and one region that covers superheated water table.
Temperatures (and pressures) are only constant and not independent of each other in the saturated water region (the area enclosed by what is sometimes referred to as the steam dome, or the saturated water and vapor curves). In the T-v diagram, these are horizontal lines of constant pressure. In the P-v diagram, these are horizontal lines of constant temperature. Outside of the saturated water region temperature and pressure are independent variables.
The reason T-v diagrams are generally used instead of P-v diagrams, is probably because the first column in most steam tables (saturated and superheated) is the temperature, and not the pressure.
Hope this helps.