If the majority of the People vote ‘no’ on 25 May 2018 there will be no change in the law.
The Constitution will continue to allow abortion only where a pregnant person’s life is at real and substantial risk (see explanation here).
The Protection of Life During Pregnancy Act 2013 will almost certainly remain in force, unchanged (see explanation here).
It will not be possible to make abortion available in situations of rape, incest, ‘fatal foetal anomaly’, serious risk to health, or risk to health. Any attempt to do so would be unconstitutional (see explanation here).