The problem being solved
Rare earth elements substitute for one another freely, so a structure can occur with cerium dominant, lanthanum dominant or yttrium dominant while remaining otherwise identical.
Treating those as one species hides real chemistry. Treating them as unrelated names hides the relationship.
The convention
The suffix in brackets names the dominant element at the site in question. Monazite-(Ce) is cerium dominant, monazite-(La) lanthanum dominant.
The root name identifies the structure; the suffix identifies which element won the competition for that site.
What it means for a collection
A label reading monazite alone is incomplete, and the suffix usually cannot be determined without analysis.
Recording it as monazite, suffix undetermined is accurate. Guessing the suffix is not.
Searching and sorting
Catalogue software and databases sort suffixed names inconsistently: some group them under the root, some treat each as unrelated. Knowing which behaviour you are dealing with saves a great deal of confusion when a species appears to be missing.
When searching, try the root name alone first. If nothing appears, the system is probably storing the full suffixed form and expecting an exact match.