P48 and P12 are abbreviations for two forms of phantom powering, a standardized method of supplying power to professional microphones. The main advantages of this method are its compatibility with dynamic microphones and the fact that no special, additional wires are required in the microphone cable.

Schoeps was the first company ever to offer phantom-powered studio condenser microphones (CMT 20 series, 1963-64). In the decades since then, this method of powering has become so well accepted that it is a standard feature for the microphone inputs on most professional equipment.

Which microphones need phantom power?

Only microphones with built-in amplifiers require powering. This includes all condenser microphones. Most dynamic microphones (including ribbon microphones) have no built-in amplifiers and do not need to be powered.

Can phantom power damage microphones?

Phantom powering was designed to be safe for microphones that weren’t designed to use it, such as dynamic microphones. However, in unfavorable cases, e.g. faulty wiring, circumstances can occur that can damage such a microphone–in particular, plugging or unplugging it while the powering is on. Schoeps microphones are protected against such damage, but it is good general practice to connect all microphones with the power supply turned off. It is switchable for most microphone inputs in professional audio engineering, and if possible should be switched on only for microphones that need it.
Please be aware also that the compatible nature of phantom powering is one of its particular design features. Other methods of powering may harm microphones that weren’t specifically designed for them.

How does phantom powering work?

Phantom powering takes advantage of the fact that balanced transmission allows common-mode signals to be suppressed in the receiving circuit. The supply voltage is applied as a common-mode DC voltage to both signal wires. The ground connection via the cable shield serves as the return conductor. In the case of P48, 48 V is used as the supply voltage, with the negative pole to ground. For P12 this voltage is 12 Volts, generally at somewhat higher current.
To avoid short-circuiting the useful signal and to limit the maximum current, the voltage is fed via two resistors, whose value is standardized and must be identical within each channel. Fig. Phantom power supply circuit


 Fig. Phantom power supply circuit
Fig. Phantom power supply circuit

IEC 61938 standard

The technical details are regulated in the IEC 61938 standard. P48 is by far the most widely used today, but P12 still has a certain importance. Most Schoeps microphones support P12 as well as P48 because it can be advantageous in mobile applications. Unfortunately, one cannot always rely on phantom power supplies to comply fully with the standard. In some portable or wireless applications it can be difficult or expensive to meet the strict requirements of the standard and to make the full voltage and current available, e.g. when a device is powered by small, rechargeable batteries. In such cases one cannot say with certainty whether the microphone will work at all or if so, whether it will maintain its full capabilities.
Some manufacturers, including Schoeps, disclose the actual requirements for their microphones to deliver full performance. For example the CMC 1 requires 30 V and 2 mA for P48, and 11 V and 3 mA for P12. If even these reduced requirements are not met, the maximum SPL can be lowered and/or distortion can increase. If the impedance of the input is lower than the standard minimum of 1 kOhm, the performance of the microphone may also be affected adversely.
Schoeps also takes care with its products that the supply currents on both signal wires be as nearly equal as possible. A difference in the supply currents could lead to magnetizing the input transformer of a microphone preamplifier, which could increase distortion. It is really up to the manufacturers of transformer-equipped microphone preamplifiers to protect their circuits against this, since this problem can also arise due to wiring errors or incorrect operation. However, some devices lack this protection; in that case it depends on the current balance of the microphone.

Does a higher voltage than P48 make sense, or does it lead to a better result?  

Modern microphones, such as Schoeps, have a regulated power supply. Therefore, phantom power above 48V has no advantage. 
There are, or have been, microphones from other manufacturers that draw their power directly from phantom power. These are mostly old vintage models. With these, a higher phantom power would probably change something, whether for the better or worse depends on the individual case. 
These microphones could benefit in terms of dynamic range, i.e. maximum SPL, if the higher external phantom power resulted in a higher internal power supply. This does not apply to microphones with regulated power supplies, where it would only increase power loss, i.e., heat. 
With modern microphones, the topic is therefore rather esoteric.

Historical context

In the vacuum-tube era, the cables for condenser microphones were heavy, expensive and troublesome. They required separate internal wires for powering, with different connectors and wiring depending on the type of the microphone. Dynamic microphones, on the other hand, all required just two conductors and a shield. When transistorized condenser microphones were introduced in the early 1960s, it became possible for condenser microphones to use the more flexible and reliable cables that dynamic microphones used, with power being delivered through the same wires that simultaneously carried the audio signals.

There were two main ways to power such microphones. One was 12-Volt parallel (a/k/a "T") powering, which was adopted by Nagra and Sennheiser early on, and thus continued to be used by film sound recordists for quite some time. But studio engineers typically use larger numbers and more different types of microphones, and they soon came to favor phantom powering because of its safety and compatibility. At that time, condenser microphones were more expensive and fragile than they generally are today, so they were often reserved for special occasions such as recording a prestigious soloist. Dynamic microphones were used for much if not most day-to-day work, and phantom powering is generally safe for them, whereas "T" powering can easily damage them (particularly ribbon microphones). In addition, when working with multiple microphones and sound sources, phantom powering offers better inherent freedom from “crosstalk” among the different microphone signals.

The French broadcasting system adopted a 10-12 Volt phantom powering method with the positive pole grounded, and Schoeps, their principal supplier of condenser microphones, started making microphones for this form of powering in 1963-64, the CMT 20 series. These microphones used an innovative, patented radio-frequency bridge circuit and transformerless output circuitry designed by Dr. Willi Küsters, the co-founder of Schoeps. They were distributed by Telefunken, Siemens and Philips.

The CMT 20 series was followed in 1964 by the CMT 200 series with improved sensitivity and the first use of the “Schoeps output circuit” that has since been imitated by countless other manufacturers. Numerous microphones of these two series were sold before 1966, when the first condenser microphones operating at the present-day standard of 48 Volts were introduced by Neumann. That voltage (rather than the 60 Volts that the capsules were designed for, and which would have given the microphones wider dynamic range) was chosen mainly to suit an early customer who happened to have it available in their auxiliary electrical system.

What established 48-Volt phantom powering as a de facto standard, which then led to its becoming a DIN standard alongside parallel powering and 12-Volt phantom, was its adoption by the German broadcasting organizations. At the time they were still publicly controlled and centrally administered as to their acquisition of equipment. Their microphone power supplies were standardized around the Telefunken AC 701 tube, and a small current from the 120 Volt plate supply could be provided to a transistorized microphone by way of a simple voltage divider at a lower voltage such as 48. Such arrangements eased the transition from tube microphones by allowing for years of overlapping use.

It's fair to say that for nearly everyone else, though, 48-Volt powering caused trouble and expense for years afterward. Most solid-state audio equipment didn't use such high DC voltages internally, so the early users of these microphones often had to buy special, outboard power supplies until consoles, recorders and preamps started to appear with this awkward, new voltage built in.

When reliable miniature DC/DC converters became available, microphone manufacturers started to offer improved models that polarized their capsules at their full design voltage for optimal dynamic range (although this then increased the required operating current, which not all consoles and recorders could supply). And transformerless output stages (also pioneered by Schoeps), with their lower output impedance and sound “coloration”, better ability to drive long cables and greater maximum SPL capability, began to be widely used.