A very early tradition of preaching in the Wesleyan / Methodist churches was for a Lay Preacher to be appointed to lead services of worship and preach in a group (called a 'circuit') of meeting places or churches. The lay preacher walked or rode on horseback in a prescribed circuit of the preaching places according to an agreed pattern and timing, and people came to the meetings.
After the appointment of ministers and pastors, this lay preaching tradition continued with Local Preachers being appointed by individual churches, and in turn approved and invited by nearby churches, as an adjunct to the minister or during their planned absences.
In addition to being appointed by members of their local churches, Local and Certified Lay Speakers of the United Methodist Church (more commonly in the United States) attend a series of training sessions. These training sessions prepare the individual to become a leader within the church. All individuals who are full members of the church are laity, but some go on to become Lay Speakers. Some preachers get their start as Lay Speakers.
In the Uniting Church in Australia, that was constituted in part from the Methodist Church, persons can be appointed:
by the congregation as a Lay Preacher; and/or
by the regional Presbytery to conduct Communion.
A well-known lay preacher was the late King Taufa'ahau Tupou IV of Tonga.
The comparable term in the Anglican and Episcopal churches is Lay Reader.
Layman, ‘laity': In short: "laity" means "common people". The English word "laity" comes from the Greek laikos which meant "of the people", "common" (common, in the meaning "unholy", "unclean" and similar). The related verb laikoô meant "to make common", "to desecrate". Whoever calls people "laity" (or "layman" or "laymen" in within the religious context), is actually calling them "common", that is, unholy or "unclean". However, most people are perhaps unaware (we pray) of the true meaning, and it has become a custom to call non-clergy, lay.