Learn Russian: Там есть стол, а стула нет.

This is the main rule of the Genitive of negation. When you say something is not present using нет, the missing object must take the Genitive case. For the masculine noun стул, adding an 'a' makes it стула.

Why did we use а for 'but' instead of но?

In Russian, а is used to show a simple contrast between two things, like 'whereas' or a soft 'but'. Here, we are contrasting the presence of the table with the absence of the chair. The word но is usually used for a stronger contradiction or an unexpected twist.

Why is the word есть used for the table, but not for the chair?

The word есть means 'exists' or 'there is'. So, Там есть стол means 'There is a table there'. In the second half, the word нет already means 'there is no' or 'does not exist'. You do not need to use есть with it because нет does the job of both the verb and the negative word.

Because стула ends in 'a', does it act like a feminine word here?

No, its gender does not change! It is very common for beginners to confuse a masculine word in the Genitive case (which often ends in 'a') with a feminine word in its basic dictionary form (which also usually ends in 'a'). It is still the masculine word стул, just wearing its Genitive ending.