The Danish word kommunion means the Holy Communion.
Today children can receive the Holy Communion, but that was not always the case in the Church of Denmark. From 1736 (when confirmation was reintroduced) through the 1800s, children could not receive the Holy Communion until they had been confirmed. In fact, young people had to receive the Holy Communion on the Sunday following their confirmation.
Some books of communions (kommunionbøger) have been kept, listing the parishioners who received the Holy Communion. If a parish register does not list confirmations or the parish register is lost and a book of communions is extant, the information about the communion can be used to establish when a young person had been confirmed - and to place the family in the parish, because a family often received the Holy Communion together.
It was not customary to receive the Holy Communion every Sunday, but only a couple of times per year. Experience tells me that engaged couples received the Holy Communion shortly before the marriage, so a book of communions can provide various clues for genealogists.
The image at the top: Arnold Mikkelsen, photograph of a communion cup and plate made by Jacob Lund about 1740; image copy, Nationalmuseet, Nationalmuseets samlinger (https://samlinger.natmus.dk/dmr/asset/214911), license: CC-BY-SA.