I'll try to touch on some basics, and give you an idea of what direct to at least ask smart questions about any particulars. I remember being overwhelmed myself the first time I wanted to order some hookup wire from an electronics supplier, rather than use whatever scraps I could find in my parts collection (aka "junk box").
The first basic property of cooper wire (I'm going to exclude aluminum and high resistance wires) is the diameter or
gauge. The two gauge standards that I am aware of are American Wire Gauge (
AWG) and (British) Standard Wire Gauge (
SWG).
From this you can use reference tables to determine the length per weight/mass (i.e. ft/lbs or m/kg), the
maximum current carrying capacity (continuous or intermittent duty) for a given max temperature (say 100C / 212F) and maximum ambient temperature (say 57C / 134F). This may be specified by the National Electrical Code (NEC) or other local building / electrical standards depending on application or regional requirements.
The maximum safe voltage potential if determined by the insulation type and thickness. This may 300, 600, 1000V as common examples. Otherwise the insulation type is important for physical characteristics, such as fire/flame resistant, toxicity (directly and/or from being heated or burnt), and flexibility (tight bends).
Bare copper wire is uninsulated, and may be pre-tinned as copper easily oxidized when exposure to air and moisture. Bare wire may be used as a short jumper, where being inadvertently shorted is unlikely or impossible and not a safety hazard. The only other application I can think of bare wire is as a sensor (moisture) or electrode. Otherwise it is not worth the small cost savings of not being coated in insulation.
The pre-tinned is being coated in tin metal or alloy (e.g. tin/lead) which improves solderability and reduces oxidation when exposed to air. This oxidation acts as an thin layer of insulator or forms a very poor rectifier (diode) producing unreliable and possibly confusing results.
Cable means simply a bundle or more than a single wire.
Most wire is either a single solid strand of the specified gauge or a twisted bundle of smaller diameter wires, so as to be easier to bend, and if a couple strands break it doesn't imply the entire bundle will break creating a
open in the wire.
Wires and cables can able be shielded (screened) with a outer conductive layer on top of the first layer of insulation of either fine braided mesh or foil (typically aluminum) and a second insulation layer.
For normal voltage and low to modest current and power requirements, most electronic projects use a reasonable fine gauge of insulated wire referred to as "hook-up wire" in multiple colours (red, black, green, yellow, blue, and white are probably most common due to colour coding conventions). I would first select four colours in 22 AWG of solid wire with whatever insulation. Next I would add 3 or 4 colours of stranded 24 AWG hook-up for connections that will be flexed more.
Then I would add 3 or 4 colours of solid 18 AWG hook-up wire, and possibly a red/black pair of "zip-wire" (two insulated conductors in parallel, also called lamp cord) for power hook-ups of solid or stranded 18 or 16 AWG.
Over time you will add wire & cables based on application needs, such as
ribbon cable for parallel data or signal connections, coax cable for radio frequency or very high speed serial usage, and magnet wire for winding your own transformers and inductors. For high-voltage applications (e.g telsa coils) you'll likely need to purchase wire with suitable high voltage insulation, and the same for power applications like power amplifiers and power supplies.
Note:
robrenz and
ejeffrey make some very good points regarding flexibility and magnet wire
Important note about wire/cables
pricing: Since approximately 2004 [1] wire and cable prices have been quite unstable as the commodity pricing of copper (metal) was quite volatile due to (real or perceived) supply shortages. Thus pricing can appear to be nonsensical, and old catalogs are useless as a price guide.
[1] I picked this year at random, but it's based on a random possibly factual graph from Wikipedia I found on the Internet.
Ref:
http://en.wikipedia.org/wiki/File:Copper_Price_History_USD.png as of 30 May 2012