Facsimile of the Autograph
These are hardcover, research-quality reproductions of the original scores from the composer. While an Urtext edition will get you close to understand exactly what the composer wrote, nothing beats studying their manuscript.
Some publishers (particularly Henle) print a hardbound, linen-covered version in addition to the standard paperbound. The music inside is the same. These editions are absolutely lovely, though not cheap.
When publishers offer multiple bindings (e.g. hardcover) or study scores, this is the "standard" version. If you're planning to play the music, this is probably what you want.
For ensemble music, this indicates that there are separate parts for each player.
For chamber pieces, playing scores have all of the parts on one system. There are not separate parts for each player. You need to memorize, share a stand, or purchase multiple copies.
For ensemble music, this indicates that the edition contains all parts on a single system (there are not separate parts for each player).
Solo Part with Reduction
For solo pieces with orchestra, this is a version that contains a piano reduction of the orchestra parts. For piano pieces, two copies are typically needed for performance.
A small (think choral size) copy of the complete score meant for studying, and not playing. They make great add ons when learning concertos and small chamber works.
In addition to the printed music, the edition contains recordings of the pieces. This may be an included CD, or access to files on the internet.
With / Without Fingering
Some piano editions (particularly of Mozart and Bach) print two copies - a pure Urtext edition that includes no fingering suggestions and a lightly edited version that includes a minimal number of fingerings. It's just an issue of personal preference.