Details vary depending on the TLD but, roughly speaking, the process (after the transfer has been requested and paid for and the requestor has supplied the authorization key to the gaining registrar) is something like this:
(1) Authorization key has been validated by the registry
(2) email has been sent to the losing account holder
(3) the transfer has been approved by the losing account holder
(4) email has been sent to the gaining account holder
(5) the transfer has been approved by the gaining account holder
(6) final approval of the transfer has been received from the registry
(7) the gaining registrar has added the domain to the account of the gaining account holder.
If losing and gaining registrars are both using state of the art technology, the whole process can easily be completed in 15 minutes.
Progress through these 7 steps is best tracked by the gaining registrar. My favourite registrar creates a table showing each of these steps, where each step is marked with a red cross, which changes to a green tick as soon as that step is completed, and with a clickable button on each step so the requestor can ask for that step to be re-tried if it seems that something got stuck.