When you intercede, you plead with one person on behalf of someone else, so 'intercession' is the formal word used for prayers made on behalf of people in need.
Intercessions can be part of someone's private prayers, but they are also a regular part of church services. When used in a service, the intercessions are spoken aloud by one person (a minister, or a lay person), as the leader and representative of everyone present. At the end of each prayer or group of prayers the person leading the intercessions will say a particular phrase (the 'bidding') and the rest of the congregation will show their consent and agreement by saying another phrase (the 'response'). A common bidding is 'Lord, in your mercy', to which the response is 'Hear our prayer', but other phrases may be used.
Sometimes intercession leaders use published prayers, but more often the prayers are specially written for the occasion. On this site you can see a number of examples of intercessions.
The intercessions often finish with this bidding:
accept these prayers for the sake of your Son,
our Saviour, Jesus Christ.
The Lord's Prayer
The Lord's Prayer is so well known that we often rattle it off without thinking about its meaning, but Jesus gave it to his disciples, and to us, to teach us how to pray. You may find our booklet on The Lord's Prayer helpful.