I’m not sure where you’re from, so the term can have multiple meanings.
- The Solemn Communion (which may be termed quasi-Solemn to distinguish it from the Solemn Communion) . It is also known as the “General Communion”, although this term may be used for situations without solemnity
- The Solemn Communion in the full and proper meaning of the term
(1) is what Lily described and what many people might be familiar with today. All the children of a particular church receiving Holy Communion on one day, with some amount of solemnity and extras like white clothes, possibly even a procession holding candles or something, etc. This usually happens annually for a parish (at least it does in my parish)
(2) The “Solemn Communion” was an institution that was very common in certain parts of Europe - it was particularly however, associated with France. I don’t know about the USA, but I imagine that possibly immigrant parishes may have retained their own traditions.
Before the decree of St. Pius X, Quam Singulari, urging an earlier age for the reception of First Holy Communion, the Sacrament was often administered at an older age - around 12 -14. Thus the Solemn Communion was the First Holy Communion . In France, it was invested with some aspects of solemnity - the children renewed the baptismal promises, were consecrated to the BVM at the end, and often ivested with the scapular; Confirmation was also received at this time and it was generally a Big Bash
Quam Singulari was the first of a series of instructions in which children were ordered to be admitted to the Holy Communion at a younger age. Consequently, this annoyed many people who had grown used to the time-honoured ceremonies of the Solemn Communion, some of which could not be done with children as young as 7.
However, the directive could not be ignored and so the practice was usually as follows. The children were admitted either to a “private First Holy Communion” which was a very “quiet” affair, or a “General First Holy Communion” every year, which did not have much, if any, solemnity.
At an older age (11-13), they made their Solemn Communion. It was no longer the First Communion, but it retained all the trappings - Solemn High Mass, renewal of baptismal promises, consecration to the BVM, or the Sacred Heart, etc. processions with clergy and sodalities, and the sacrament of Confirmation. Accompanied of course, by the non-liturgical bits like new clothes, gifts, parties, and so forth.