Critical Edition - Also known as the Collected or Complete Works, Gesamptausgaben, or Opera Omnia. They are based on scholarly evaluation and collation of sources, taking into account variant readings and innumerable aspects of contemporary performance practice. They often come with extensive notes, sometimes in separate volumes, showing how decisions were arrived at.
Facsimile - A photographic reproduction of the composer's manuscript, or an early copy, or a first published edition.
Performing Edition - Meant to be used during practice and performance. A performance edition often aims to meet the performer's practical needs, such as including helpful fingerings, providing a layout that facilitates page turns, and suggestions for phrasing. Some performance editions are edited by well-known performers whose suggestions might be valued by a pupil.
Public Domain - Music that is no longer covered by copyright. In the US, any music published prior to 1923 is considered in the public domain, but the laws vary in different countries. For more information on Copyright & Public Domain music, take a look at the Music Library's Copyright Page where you'll find some details on what Fair Use is, as well as some clever videos.
Reprint Edition - A reprint of a previous edition. Sometimes it does released by the original publisher, but most often by a publisher like Dover or Kalmus. They essentially scans an old edition and bind it with no editorial changes, including leaving clear errors that were made in the original they are working with.
Thematic Catalog - An index to a group of musical compositions - most often those of a single composer - that incorporates citations of their opening notes (incipits), or principal melodic features (themes), or both. These citations may be given in various forms, such as conventional notes, neumes, tablatures, syllables, numbers, letters or computer codes.
Urtext - Based on scholarly work that is intended to embody the composer's original intent with minimal editorial interference. These are frequently released by the same publisher responsible for the Critical Edition if there is one, and are often used by professional musicians as their performing edition because they are a blank slate for the performer to make their own editorial/performance decisions.