Each atom of iron has an unpaired electron whose spin can be lined up to that of the unpaired electron from a neighboring iron atom. The spinning of the charged electron creates a magnetic moment, which in turn can align with an external magnet, thus making iron magnetic. Atoms of wood do not have unpaired electron spins that can line up with a magnet, and so it is non-magnetic.
A charge that is spinning has a magnetic moment which can be influenced by neighboring charges or an external magnet. Iron atoms have unpaired electrons whose spins align with the unpaired electrons of the neighboring atoms. When the spins of a small region of iron line up with one another, that region acts like a magnet and is called a domain. Naturally occurring iron itself does not form magnets, because a piece of iron contains several domains and the spins of the electrons in each domain work against the others, thus canceling out any net magnetic effect. If a piece of iron is brought close to an external magnet, the spins of all domains line up with the direction of the magnetic field of the external magnet, thus changing the piece of iron to a temporary magnet. Even after the external magnet is moved away, the iron remains magnetized for a short period of time because of the alignment of the domains.
Wood, on the other hand, does not have free unpaired electrons whose spins can line up to form domains. The magnetic moment of the individual electrons is not available to be aligned with an external magnetic field. This is the reason wood is not attracted to magnets and cannot be magnetized.