What is mastering?

Mastering is both the last stage of production and the first stage of manufacturing. It is the last chance to alter the sonic presentation and is where the various parts of an album are brought together into a cohesive whole.

The mastering room provides a different (hopefully accurate) monitoring situation and a set of fresh (hopefully experienced) ears to help the artist and producer achieve their vision for the final record.

Some of the processes that might be involved:

Editing is often done in the mastering room. I have often been asked to make long songs shorter and to make short songs longer. On other occasions, I've had to fix a bad note, for example in a classical piano performance. Sometimes edits are needed to remove clicks and other spurious noises.
Equalization, an audiophile "no no" but in fact this often helps when you consider the monitoring situation in most studios, common microphones and the techniques employed in their use. The best recordings (the minority to be sure) don't need this.
Level adjustments to ensure songs flow consistently from one to the next, without the listener having to constantly adjust their volume control.
Sequencing (i.e. song order) and spacing between tunes.

Many will add to these things like compression and/or (for digital recordings) "normalization".

Personally, while I can see compression as an effect on individual tracks of a multitrack (e.g. to get Ringo's cymbal sound), I don't at all like its use on whole mixes where it is generally used to achieve more loudness (some say "punch" but how do you increase punch by taking away dynamics, where the punch lives?).

This is a subject about which much can be and has been written. Many mastering engineers today "compete" on the basis of how loud they make records. A good number of record producers and A&R folks still seem to think we buy records because they are loud and not because we like the music. (Go figure.) From my perspective, all the best sounding recordings I have heard have in common the fact that they are not loud. Having loudness as a goal necessitates the sacrifice of dynamics which is something I'd prefer not to do.

"Normalization" (I call it abnormalization) is a means of raising the level of a digital recording so the peaks reach maximum level. This is a trap many semi-pros (among others) fall into. When done for an album project, it results in "quiet" tunes that sound louder than the "loud" tunes. In addition, when done to a 16 bit recording, as is often the case, this results in truncation of the low order bits (which carry a lot of important musical information), adding harshness and obliterating the stereo image by losing important spatial cues.

So in sum, mastering is where the "part" used for replication is created. It is the final step in the creative process.