If I am not mistaken, infant communion is having small children and infants participate in the Lord's supper, even without a profession of faith, so long as they've been baptized. Is this a proper understanding of the practice? What is the history behind it?
Yes to the first question. With regard to the second, the practice was widespread in Christendom until the High Middle Ages, roughly speaking. (If I recall correctly this roughly coincided with the Roman Catholics beginning to withhold the cup from the laity for similar reasons.)
The Eastern Orthodox bodies continue to practice paedocommunion to this day. Many sources assert that the practice was well nigh universal in the Western church as well until the Romanists abandoned it. But Rev. Matthew Winzer wrote an article a few years ago in which he argued that the practice was not universal.
Is this practiced in your church?
No, but of course I'm a Baptist.
I doubt it's practiced in any of the member's churches, at least not the regular posters. In "Reformed" circles (speaking very broadly to include nonconfessional churches that nonetheless fly the Reformed banner) this practice is largely confined to those organizations that embrace Federal Vision theology. That theology has been rejected by every major confessional Reformed denomination in North America. (And probably many elsewhere as well.) As noted in the board rules, advocacy of Federal Vision theology (FV) is forbidden on the Puritan Board.
What do the confessions say about it?
The practice is precluded by the confessions and by Scripture. Off the top of my head I can't recall if any of the confessions directly address the issue in so many words, but the practice is forbidden due to the necessity of self-examination etc. that is noted in the confessions and/or catechisms. There have been some Study Committees in some denominations (the OPC comes to mind) that predated the FV controversy. However the rejection of paedocommunion was upheld.
Given the recent FV controversy, there is a lot of information about paedocommunion online. Monergism has a lot of it linked. There have been a good number of threads on this board addressing it as well.
