  The electronic structure of Ba$_2$Ti$_2$Fe$_2$As$_4$O, a newly discovered
superconductor, is investigated using first-principles calculations based on
local density approximations. Multiple Fermi surface sheets originating from
Ti-3$d$ and Fe-3$d$ states are present corresponding to the conducting
Ti$_2$As$_2$O and Fe$_2$As$_2$ layers respectively. Compared with
BaFe$_2$As$_2$, sizeable changes in the related Fermi surface sheets indicate
significant electron transfer (about 0.12$e$) from Ti to Fe, which suppresses
the stripe-like antiferromagnetism at the Fe sites and simultaneously induces
superconductivity. Our calculations also suggest that an additional
N\'{e}el-type antiferromagnetic instability at the Ti sites is relatively
robust against the electron transfer, which accounts for the anomaly at 125 K
in the superconducting Ba$_2$Ti$_2$Fe$_2$As$_4$O.
