Yes, they turn into something new -- another kind of atom, one with a different number of protons, or neutrons, or both.
Recall that half-life is not a process, but a time interval -- the time required for one half of the atoms in a radioactive sample to undergo decay. When one half-life has elapsed, half of the original atoms are still there, unchanged (yet). They are simply the ones whose random chance of decay has not yet come up. After another half-life, one quarter of the original atoms will remain unchanged, and so on -- one eighth, one sixteenth....
Just which atoms will have decayed during any interval is entirely unpredictable, however, and the half-life of a given kind of radionuclide cannot be changed.
A daughter nuclide is the product of the radioactive decay of another atom's nucleus (a nuclide). The term 'daughter' is used because, often, such a nuclide is still unstable and can give rise to (analogous to "giving birth to") yet another kind of nucleus.