When the first element of the periodic table is treated with dioxygen, it gives a compound whose solid state floats on its liquid state. This compound has an ability to act as an acid as well as a base. What products will be formed when this compound undergoes autoionisation?

The first element of the periodic table is hydrogen and its molecular form is dihydrogen (H2). When H2 reacts with O2, water is formed. Water is a liquid at room temperature. When liquid water freezes, it expands to form ice.
Density of ice is lower than that of liquid water and hence ice floats over water. Water is amphoteric in nature. It acts as a base in presence of strong acids and as an acid in presence of strong bases.
$\underset{{\mathrm{Base}}_{1}}{{\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{l}\right)}+\underset{{\mathrm{Acid}}_{2}}{{\mathrm{H}}_{2}\mathrm{S}\left(\mathrm{aq}\right)}\underset{{\mathrm{Acid}}_{1}}{\to {\mathrm{H}}_{3}{\mathrm{O}}^{+}\left(\mathrm{aq}\right)}+\underset{{\mathrm{Base}}_{2}}{{\mathrm{HS}}^{-}\left(\mathrm{aq}\right)}\phantom{\rule{0ex}{0ex}}\underset{{\mathrm{Acid}}_{1}}{{\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{l}\right)}+\underset{{\mathrm{Base}}_{2}}{{\mathrm{NH}}_{3}\left(\mathrm{aq}\right)}\to \underset{{\mathrm{Acid}}_{2}}{{\mathrm{NH}}_{4}^{+}\left(\mathrm{aq}\right)}+\underset{{\mathrm{Base}}_{1}}{{\mathrm{OH}}^{-}\left(\mathrm{aq}\right)}$
Due to amphoteric character, water undergoes self ionisation as shown below
$\underset{{\mathrm{Acid}}_{1}}{{\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{l}\right)}+\underset{{\mathrm{Base}}_{2}}{{\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{aq}\right)}⇌\underset{{\mathrm{Acid}}_{2}\phantom{\rule{0ex}{0ex}}\left(\mathrm{Conjugate}\phantom{\rule{0ex}{0ex}}\mathrm{acid}\right)}{{\mathrm{H}}_{3}{\mathrm{O}}^{+}\left(\mathrm{aq}\right)}+\underset{{\mathrm{Base}}_{1}\phantom{\rule{0ex}{0ex}}\left(\mathrm{Conjugate}\phantom{\rule{0ex}{0ex}}\mathrm{base}\right)}{{\mathrm{OH}}^{-}\left(\mathrm{aq}\right)}$
This self ionisation of water is called auto-protolysis and autoionisation.