Why do we bow in Aikido?

The bowing is meant as a mark of respect to  the instructor or your partner - it does not have any religious significance and is in fact more akin to a handshake in Western society. It does not symbolise worship of any sort.

Another reason for bowing is as a safety measure - a physical pause that is used to put aside extraneous thoughts and consider the limitations of your training partner (among other things).