A non runner is a horse that was scheduled to start a race but has since been removed from the starting line-up. This can happen for a variety of reasons, including injury.
Normally, any bets that have been placed on a non runner will be voided and you will receive your stake back. If the non runner is part of an accumulator (multiple) bet then this bet is normally removed from the accumulator and it continues as normal (i.e. a four-fold accumulator would become a treble).
The exception to these rules is a bet that had been placed as an Ante-Post bet, in this situation the bet is normally settled as a loser.
PLEASE NOTE: These are general rules, you should always check the exact rules for the bookmaker where you have placed your bet to determine if they are different.