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What kind of material can be cut?

Any material that conducts electricity, regardless of hardness.
Aluminum, magnesium, copper, brass, titanium, steel of any type/alloy/hardness, carbide, gold, silver, platinum, graphite, and many others.

How fast is the wire edm cutting process?

Typically, the wire edm process is slower in material removal than other "conventional" machining processes. However, this does not necessarily mean that completing a part is slow.
There are times when even though a part can be made using conventional machining methods, it can be made faster with wire edm. An example of this might be when a thin, delicate part has to be made from a large block of material due to its configuration -- conventional machining would require all the excess material be machined away, whereas wire edm might be used to drop out all the excess material with one or two simple cuts.
Direct comparison between conventional machining methods and wire edm can not be "generalized" however, and must be looked at on a case by case basis.

What determines cutting speed?

Cutting speed is primarily determined by...
1) Size of wire used
2) Thickness of the part
3) Density of the material
4) Profile/shape being cut

What is the range of wire sizes used?

Wire size ranges from .0008" (20 micron) to .013". Most wire edm machines are designed to run .006" to .010" diameter wire. My Agie will run from .0028" (70 micron) to .013" wire. Machines using .0008" wire are very specialized, and typically have a very small "work area" designed primarily for tiny parts used in the medical or electronics industry.

What determines the size wire used?

Typically, wire size is determined by the smallest inside radius required on the part. For example; a part might contain a square hole with a maximum radius of .003" in the corners. In this case, the choice would probably be .004" wire. Generally speaking, it's advantageous to use the largest wire possible as that will cut the fastest (large wire can carry more power proportionally to the amount of material being cut away). There are occasionally situations where using smaller wire is beneficial though... primarily when cutting thin material with intricate/fine details.

How thick a part can be cut?

Most wire edm machines have a "Z" capacity in the 10" to 12" range. Larger capacity machines may be able to handle parts as thick as 16" or more, and very specialized machines have been built that will cut parts 24" thick or more. On the opposite end of the scale, ultra-fine wire machines using 20 micron wire may only have a Z capacity of 2 or 3 inches.

How thin a part can be cut?

Cutting .001" or even .0005" thick material is simple. When cutting very thin parts, the material is usually "captured" between two sacrificial plates, and the entire "stack" is cut as a unit.

How accurate is the wire edm cutting process?

More accurate than any conventional machining process.
Wire edm has one distinct advantage over any form of conventional machining -- there are no cutting forces. The wire never contacts the workpiece, so the "path" that the wire takes through the part is not influenced by any mechanical forces. Additionally, wire cutting is typically done submerged in very pure, temperature-controlled water, so dimensional changes due to temperature are minimized or completely eliminated. An extra benefit to having no cutting forces is that parts can be held very lightly, which minimizes or completely eliminates the distortion due to clamping forces seen in conventional machining methods. Thin/delicate parts can be cut very precisely using wire edm.
In order to achieve very high accuracy though, it is often necessary to make more than "one pass" on a profile.

Why is more than one cut ever required?

Two reasons; greater accuracy and/or finer surface finish (the two go hand-in-hand). When cutting using a "single pass", power settings are usually quite high and consequently the surface finish will have a "glass-beaded" texture to it after cutting. Additionally, two other factors can slightly influence wire position during the first pass of a wire cut -- electromagnetic forces on the wire, and high "flushing pressure" used to continually wash away the eroded material. These forces do not move the wire very much (a few tenths possibly), but this neverthess affects accuracy.
In order to achieve greater accuracy, multiple cuts are employed where the "offset" of the wire is reduced (easier to think of it as "spark gap"), and the electrical settings are altered (lower power and different waveform). When very fine finishes are required (Ra10 / 10 micro-inches or better ), as many as 4 or 5 additional "trim passes" can be used after the initial cut. Each pass will have less offset and lower power than the previous one. Whether it's fine finishes or ultra-precise tolerances that are required, multiple passes are the way it's done in wire edm.

What is the biggest limitation of wire edm?

The single biggest limitation of wire edm is that it must be possible to pass the wire by or through the part. The wire is continuous... so it's not possible to cut a "blind" cavity using wire edm.

Can complex shapes be cut?

Yes, within some reasonable limits. Most modern wire edm machines can cut up to 30° taper (or more precisely; the wire can be inclined up to 30°). Note that since the upper axis on a wire edm machine is typically a "differential" axis (it translates relative to X/Y), a 30° angle is possible up to a certain part thickness, and after that the available angle reduces relative to the height of the part. Obviously if the part itself can be tipped or rotated, any angle is then possible.
Additionally, the angle of the wire can be continuously changed while cutting, which allows the cutting of complex surfaces. Essentially; if a straight line can be passed along a contour, it can probably be cut by wire edm. To see a simple yet illustrative example, take a look at the twisted cube. Note that there are surfaces that would "appear" impossible to cut with a straight wire, yet this was easily done. As you can see; "morphing" a shape on one end of a part into a different shape on the other end of the part is possible.

How accurately can you cut a feature on my existing parts?

The key words in that question are "existing parts". Since an existing part is often fixtured on a previously machined surface (or surfaces), the limitation in accuracy is often those already-machined surfaces. If a particular feature is cut via wire edm, that feature will have all the accuracy that the wire edm machine is capable of, but positional accuracy may be dependent on previous work. In a one-off situation, this issue can sometimes be negated by "touching off" on all previously machined surfaces and striking the best possible balance of dimensions to achieve positional accuracy. In production however, fixtures may be used that rely on previous machining to mount/clamp parts quickly, and if those parts are not consistent, accuracy may suffer.
Wire edm is really no different than any other form of machining in this regard, except that wire edm machines are typically much more accurate than other types of CNC machines, so small variations in accuracy from part to part become obvious and glaring errors when compared to what is possible in the wire edm machine.

How do I design a part to take advantage of wire edm?