The winter 2007 issue of Montres magazine has what I think of a very good article on finishes and to exactly what they do.
This information I think goes very well hand-in-hand with the Geneva Seal information posted earlier by Alex.
So as not to infringe on copyright I will just write a few things:
Note: since the articles contain very technical words in French and so as not to confuse everybody by my lack of knowledge on what the english versions are, some of the terms have not been translated. I hope you find this useful.
1. Anglage ou Chanfreinage (beveling I think):
Anglage is used to remove the little bits sticking out from machining work on the edges of pieces and holes. These could potentially damage other parts when the movement is running and/or break off and cause damage or stop the timepiece. Bridges also need chanfreinage so as not to damage "platines" during assenbly. Polished/bevelled screw heads also reduce the risks of damage during assembly while manipulating with the screw driver. Anglage also helps against corrosion. It is usually done by hand and requires several days of work. 45 degrees angles are desired. Image below is from "forumamontres".
Polishing of the pieces is essential as the good spread of oils in the movement is largely dependant on the quality of the polishing of the pieces. All the pieces that work in contact with another one of on top/below another one need to be polished to a high degree. Examples: the gear teeth need to be polished to have achieve better efficiency and therefore ultimately more power reserve. All pieces of the movement benefit from polishing as it also increases the strentgh of steel. Three types of polishign exist: circular for round pieces, spiral for main spring barrel, and "nervures" for the 3/4 plates.
The best level of polish is the mirror polish which makes pieces look like mirrors; this also helps against oxydizing of the pieces. This is only doen for very high watches and parts (example: hammers for repeating watches).
3. In the past, when watches were not fully humidity and dust proof, all the steel parts (and especially the screws) were polished to a high finish to help fight against rust as water and dust stick a lot less on smooth surfaces. On the contrary, non-steel parts around the movment (bridges and plates) wwre made rougher on purpose (eg: with Cotes de Geneve decorations) in order to 'keep' or 'trap' the dust so that it did not fall into the watch movement itself.
4. Blue screws:
In the past, before technical advances in materials, steel screws were heated to strengthen them, giving them a blue color. Although this is not really needed nowadays, it is still difficult to obtain a consistent blue color for the screws.