Screwdrivers are one example of a type of tool called a "driver" which is simply a device with a shaft emerging from a handle, at the end of which is a tip designed to spin some type of fastener which is tightened by the application of rotation. As their name implies, screwdrivers are meant to drive screws.
As there are many different types of screw, there are many different types of screwdriver. The primary defining characteristics of a screw are the length, thickness, and the type of thread, but the primary defining characteristic of a screwdriver is its tip.
By far the best known screwdriver tip is the flat-blade tip. It is flat and slightly tapered, somewhat like a chisel. Flat blade screws are very rarely used in new devices but you can find them everywhere, even on electronics, if you examine older equipment or construction. Everything is wrong with this design except the ease of manufacture of both screw and driver; The straight shape means that in most you must be looking at the screw to place the screwdriver, and the beveled design means that the driver has a tendency to "cam out" of the slot when turned. Flat-blade screwdrivers are measured across the width of the tip in either inches or millimeters.
The second best-known screwdriver tip, and entirely the most popular, is the philips. It is shaped like a cross, and has a beveled tip. This tip has a number of significant advantages over the flat-blade type. First of all, it has flat sides, so it does not tend to cam out. It also has four ends, where the flat blade type has two, so much more torque can be transmitted to the screw. Finally, the beveled design means that it is possible (in some cases) to turn a screw without having the axis of the driver lined up perfectly with that of the screw. Philips screwdrivers come in assorted sizes and are referred to by number; #0, #1, #2 and #3 are by far the most common sizes.
Closely related to the philips is the pozi-drive screw, seen most commonly in automobiles developed between the 1970s and 1990s by general motors. It has some additional square surfaces inside of the crevices of the philips tip, in order to strengthen the tip and add additional surfaces to transmit even more power. They come in most of the same sizes as the philips.
As functional as these designs are, even more useful shapes have since been cut into the heads of screws. Most people have seen a hexagonally-driven screw. A hex screwdriver (sometimes called "allen", like an allen wrench) can have a sort of hexagonal ball shape cut into their end in order to turn hex screws at an angle, but in general one must insert the tool into the screw in a completely straight fashion. SAE hex bits are available in basically every 1/64" increment between 1/64" and 1" or so, and of course in larger sizes than that, but the most common sizes are the 1/16" inches between 1/16" and 1/2" or so. Metric hex runs from 1.0 or 1.5mm up to about 12 or 14mm in general, though again, larger sizes are available.
The Torx driver, a six-pointed star-shaped design, is even better (in most ways) than the hex head drive. As its name implies, its purpose is the maximum application of torque to the screw. Either of these designs can commonly be found with a small peg in the middle of the cutout, which requires the screwdriver to have a hole in the middle of the tip - these are known as "tamper resistant" or "security" hex or torx. The bits are readily available at swap meets and from catalogs or from internet retailers. Torx drivers are numbered and do not have SAE or Metric sizes. The most common sizes are between 5 ("T5") and T20.
There are other tips, however, which are in common usage. The square-tip screwdriver (it is also known as a "robertson" tip, and Snap-On tools calls it the "Scru Lox" tip) provides the same functionality as the hex tip, but has added strength, and they are commonly used for woodworking. There are also two common types of clutch-tip screwdriver. The "Type G" clutch tip, used by general motors, resembles an hourglass with a somewhat rounded top and bottom, as if you laid a board in the snow, anchored its center, and dragged it around about forty-five degrees of travel. The "Type A" commonly used on aircraft looks like a rectangle superimposed on a circle, such that it becomes a round shape with square tabs sticking out on opposed sides of the circle.
Naturally, many other types of tips are also used, including twelve-point or "triple square" tips, triangular tips, and shapes with odd numbers of points such as seven, intended to dissuade tampering.
Design elements of the screwdriver tip can be as important as their shape. Besides being made of different materials, they can have a feature called "anti cam-out ribs", or "ACR". These are small sharp ribs on the tip which bite into the screw, helping to prevent the cam out effect. They add little to the cost of the tool and it could reasonably be considered a mistake to purchase any screwdriver without ACR.
As there are many types of screwdriver tip, there are also many types of screwdriver. The most basic type of screwdriver is only a handle and a shaft, but even a screwdriver like that can have differentiating features. For example, they can have wrench flats near the handle at the base of the shaft or shank of the tool. You can place a wrench on this (even a ratcheting one) and use it to apply torque to the screwdriver while you press down on the handle, for extremely stubborn screws or for tapping into wood. Some screwdrivers have small plastic handles designed for maximum grip, but other have large ergonomic handles usually made from plastic designed to be comfortable to grip and turn.
Offset screwdrivers resemble a screwdriver shaft with a tip on each end and bent (usually) ninety degrees about three centimeters from the end. If both ends are the same, one of them will usually be rotated forty-five degrees from the other, but they are just as likely to be different types of tips - usually philips and flat-blade. They can be used to gain access to screws in restricted locations and are often handy for disassembling parts of a whole in an order not recommended by the manufacturer.
"Miniature" screwdrivers, also referred to as "precision" or "jeweler's" screwdrivers, are exactly what the name implies - very small screwdrivers. This refers primarily to the size of the tip, though they are small in diameter to help avoid overtightening by giving you better feel for the screw. Most of them are made of metal and have a sort of wheel on top which can be braced against the base of the thumb while you spin them, making them quite convenient to use. Higher-quality tools are usually made of some sort of oil-resistant plastic. The philips screwdrivers are usually numbered #00 and #0 and the flat-blade types measured in mm width, from about 1.0 or 1.5 up to 3.0 in a set with both types.
The most common extra feature for a screwdriver is that it may have bits, which are replaceable and exchangeable tips. The tip of a screwdriver is nearly always the first portion of the tool to be damaged, and a screwdriver with a damaged tip also damages screws. Being able to replace the tips means you can keep the tool much longer, and also means that one tool can fit a broader variety of screws. Besides being able to exchange assorted screw tips, you can also attach nut driver tips or, commonly, an adapter to allow you to mount a 1/4" drive socket to the end of the tool.
Besides simply being able to change tips, there are a number of other features which can be found on modern screwdrivers. Many of them, especially those with replacable tips, are also magnetic. As most screws are made of ferrous materials, they can be held onto the end of the screw by the magnet, or picked up with it. Some screwdrivers even have a telescoping magnet inside of them, as well as flashlights, bit storage, laser pointers, and other assortments of tools. You can purchase magnets with holes down their center and slide them down the shaft of the magnet, gluing them in place if necessary, but this produces a large magnetic field around the tool which is likely to attract large quantities of iron filings, whereas magnetic screwdrivers usually have a small magnet installed in the tip.
Ratcheting screwdrivers (predictably) employ a ratchet mechanism such that you need not extract the tip of the screwdriver from the screw in between twists of the driver intended to push the screw through the material. Usually these types of screwdrivers also feature storage space for bits. There are also ratcheting offset screwdrivers, the majority of which have replacable bits.
Finally, there is the electric screwdriver, which utilizes a battery or batteries and an electric motor to drive screws. They are usually somewhat bulky and heavy but they do spare you from doing most of the work and are very useful in cases where you must drive many screws. A cordless drill (or even a corded one) will do much the same job but is much more likely to strip screw threads and is much more heavy and bulky. Some electric screwdrivers have a pivoting head which allows you to get them into relatively close spaces.
The best screwdrivers are made of somewhat flexible (mild) steel. This allows them to flex somewhat, which keeps them from breaking easily, and it is what makes your old screwdrivers such useful pry bars.
More interesting than the metal the driver itself is made of is the coating applied to it. You may notice that while the shaft is often chromed for corrosion resistance (and to make it look nice) the tip is not, because that would make it overly smooth and "slippery", and it would cam out more easily. The tip is usually either left bare, etched with an acid, anodized with an acid and electricity, or vapor blasted. While sand blasting was once common it damages the tips of tools, making them less precise, so now the tips are sometimes blasted with moisture vapor which creates a much finer texture.