In Islam, a khutba is a sermon. It's usually given before the afternoon prayer on Fridays, when Muslims congregate to pray at a mosque. It's also performed on the two holidays Eid ul-Fitr and Eid ul-Adha.

An Imam usually gives the khutba, though often a mosque will invite a notable guest as the speaker. The speaker of the khutba is known in arabic as the "khateeb." The purpose is similiar to the sermon that is read at a church on Sundays, the purpose is to increase people's faith and remind them of God. Listening to a khutba focuses a Muslim's loyalty to Allah,  with the main objective of inspiring and motivating Muslims to order and practice virtue and forbid vice. The objective of the khutba is to give the muslims hope and encouragement. Forgiveness is emphasized, and things like pointing fingers and divisive issues are usually not discussed.

The Qur'an says "O you who believe! when the call is made for prayer on Friday,  then hasten to the remembrance of Allah and leave off trading; that is better for you, if you know." (Qur'an 62:9) The remembrance, also known as dhikr, is interpreted as "While performing the salaat of Jummah." This is supported by numerous authentic Hadiths. The following appears in Sahih Muslim: Narrated Abu Huraira "The person who takes a bath then  comes  to the  Jum`a  prayer,  then offers the prayer that was destined for him, and then keeps silent till the Imam finishes the sermon, and then  prays  along  with  him, his sins between that time and the next Friday would be forgiven,  and  even  of  three  days  more" (similar  hadiths  appear  in  Abu Dawood, Ibn Majah, & Ahmad bin Hanbal).

The khutba is usually given in the following format, broken up into two parts. After the Azan is called, the khateeb (speaker) stands up and goes to the minbar (pulpit) or podium, if there is one. The sermon begins by praising Allah and then begins speaking about an Islamic topic that is illustrated by the Quran. The speaker usually quotes a few verses and hadith regarding the topic, sometimes drawing on examples in the Prophet Muhammad's(pbuh) life. It is similar to the homily read at a church. The khateeb ends the beginning with another supplication to Allah asking for His forgiveness and guidance, with the listeners replying "Ameen," and then sits down. That ends the first khutba.

After a brief moment to collect himself, the khateeb stands again. The second khutba is much shorter than the first, and usually is largely devoted to supplications and prayers to God, whereupon the listeners reply "Ameen" together. Usually a social issue is discussed, within an Islamic context that is usually relevant to the first khutba. Political events are to be mentioned, if at all, only to relate causes and effects that are relevant to Muslims, to educate and to raise their awareness. A topic that unifies the people is encouraged, and topics that cause divisions or point fingers are strongly discouraged. The khateeb then prays for blessings and prayers upon the Prophet Muhammad(pbuh) and his family, as well as Abraham(pbuh) and his family.

After that, the khateeb instructs the muezzin to give the Adhan once more. The Imam, the khateeb, or the most knowledgable person then leads the group in prayer aloud. The khutba is considered a part of the Friday prayer, therefore the group prayer that follows it is only half as long, shortened to two sets instead of the usual four.

Attending the khutba and group prayer is a religious obligation on every adult Muslim male. Women are encouraged to attend, but not obligated. This is for the practical reason that if a woman is raising children or running a household, she is encouraged to pray at home instead. Also, during the sahaba's time, a number of women were harassed in public on the way to the mosque, so the Caliph announced that it would be better if the women prayed at home, though it was not mandatory in that regard. There are numerous hadith that urge or command Muslims to attend the friday prayers and listen to the khutba. The Prophet Muhammad(pbuh) is recorded as saying, "The person who takes a bath then comes to the group prayer, then offers the prayer that was destined for him, and then keeps silent till the Imam finishes the sermon, and then prays along with him, his sins between that time and the next Friday would be forgiven, and even of three days more" (Reported in Sahih Muslim, with similar hadiths in Abu Dawood, Ibn Majah, and Ahmad bin Hanbal)

Traditionally, the khutba given by the Prophet Muhammad(pbuh) did not exceed ten minutes. His prayer was longer than his khutbas. Some of his companions took longer, but no more than about 25 minutes.

Rules and Guidelines for giving a khutba can be found at http://www.themodernreligion.com/basic/pray/prayer_jumakhutba.htm

You can listen to some khutbas online at http://khutba.org/ The text of many khutbas can be read online.