How does Borax Solution act as a buffer?
Borax reacts in hydrochloric acid in the preparation of the buffer to give the boric acid through the following reaction $\ce{Na2B4O7.5H2O + 2HCl -&gt; 4H3BO3 + 2NaCl + 5H2O}$

http://www.novedu.ru/anotes/refer/buffers.htm#bor1

The above link gives you the procedures to making the buffer. The reason why the borax solution is able to keep the system at around pH of 9 because the boric acid that is produced from this reaction has a pKa of 9.46. When the concentrations of the acid and its conjugate base, that is, the borate ions is exactly 1:1, the pH will be most difficult to change since any addition of hydrogen ions will only bond with the borate to form the undissociated boric acid. 

So to answer you question, boric acid is the weak acid, and borate ions are the conjugate base.