We should begin this answer by considering the definition of a saint. Because we are accustomed to thinking of sainthood as something attainable to only a select few, it is difficult for us to realize that we, in fact, are part of the Communion of Saints. A saint is a good person; not a perfect person but a good person. They strive to choose what is good; to be holy. Just as Jesus called his disciples to holiness, we too are all called to be holy – to be saints.
This question comes up often, usually from someone who has attended a wedding or a funeral Mass, received communion, and is planning on attending Mass again later in the day. Like many things in the Church, this law was changed in the wake of Vatican II.