A visitor who walks into a Mormon church on the first Sunday of a month may find that he has walked into a slightly different meeting than the form the normal Mormon church services take.
The first Sunday of the month, also known as Fast and Testimony meeting, is a time when members can share their testimony and they spend the day fasting in order to gain spiritual insight and knowledge. Instead of the three speakers that normally make up a Mormon service, members of the congregation can get up and bear their testimony.
The purpose of fast and testimony meeting is to give members a chance to share their faith with each other. No member is required to do this, and many Mormons choose to remain in their seats and just listen to the testimonies of the other members of the branch or ward. It is more free form than the typical Mormon meetings and while there are common elements to many testimonies, judging from the experience of the author, the testimony each person gives can very widely.
A more important part of the fast and testimony Sunday is the fast offering. The fast offering is a monetary donation given by most faithful members to help the members of the ward in need. The fast offering is supposed to be the amount the individual or family would have spent on meals for that day.
Members can become emotional in Fast and Testimony meetings and crying is not uncommon. Not every member of the congregation participates and while some Mormons bear their testimony to the members of the congregation they belong to once a month, other people never bear their testimony in public. After the closing prayer of sacrament meeting, the congregation will go to their respective Sunday school classes.
The rest of the Sunday meetings are not the same. In areas with a higher Mormon population, members of the Deacon quorum may be sent out to collect the fast offerings. In areas with a low Mormon population, fast offerings are collected in a tithing envelope.
There are two times a year when fast and testimony meeting is not held on the first Sunday of the Month. In April and October, the date for fast and testimony meeting may be moved ahead one week or back one week. The first Sunday of these months is when the church holds a general conference, which is when the leadership addresses the population. Many members skip church during this week as the general conference sessions are televised.
Resources:
Mary Joley -- Fast and Testimony Meeting