What is the LDS practice of home/visiting teaching? Are they the same thing? Where did this originate, as I don't know of other sects which have the same practice... at least with those terms.
Home and visiting teaching are assignments given to church members to visit with one another, in order to uplift and strengthen each other, and to identify needs which each members may have.
The basis for this can be found in LDS scripture, Doctrine & Covenants 20:46-47:
The priest’s duty is to preach, teach, expound, exhort, and baptize, and administer the sacrament, and visit the house of each member, and exhort them to pray vocally and in secret and attend to all family duties.
Generally, during a home teaching visit, the home teacher will talk with the family, find out what's going in their lives, and usually share some spiritual message.
Most men in the church are given this responsibility.
The difference between home and visiting teaching is that home teaching is done by the men, and they visit with the whole family, while visiting teaching is done by the women, and they only visit with other women.