24-pin power supplies and you
The 2.0 iteration of the ATX specification introduced a simple but much-noted change. It replaced the classic 20-pin "ATX" connector (what the standard calls more simply the "main connector") with a 24-pin connector.
The basic motivation behind the change was to provide more power (and better power) to increasingly power-hungry video cards. The PCI-Express standard not only allows a video card to draw more power from the PCI-Express slot, it also permits configurations with multiple video cards in one system. In typical use, these multi-card solutions (whether SLI or Crossfire) are based on higher-end cards that have a direct connections to the power supply anyway, and therefore don't need to draw much power from the slot. But it was a reasonable concern.
In systems where the video card(s) connects directly to the power supply, or systems with integrated graphics, there is little benefit. The motherboard gets a slightly better connection to the power supply, and that's it.
The big change, as stated, was in the main connector. 4 new pins were added to the existing 20-pin connector. These carry (respectively) 3.3 V, 5 V, 12 V, and ground. An exact pinout is available at pinouts.ru (pin 11, 12, 23, and 24 are the new ones).
There is still a great deal of older ATX 1.x equipment on the market and in use. The ATX 2.0 spec deals with this by ensuring good cross-compatibility with ATX 1.x parts.
Many power supply manufacturers have implemented "20+4 pin" connectors. These have the traditional 20-pin connector, and then a block of 4 more pins that can be removed. This ensures complete compatibility with both new equipment and old.
There are a few gotchas in mixing ATX 1.x and ATX 2.x parts. The latch meant to hold the main connector in place may not line up with the one on the motherboard. This is generally not an issue, because the connector fits snugly anyway, and most motherboards do not have components close to that side of the connector.
Works fine. The 20-pin supply can simply be plugged into the motherboard and will Just Work provided the motherboard doesn't attempt to draw a great deal of current. Any problems will manifest as system instability under load.
A number of companies sell adapters meant to resolve this situation; those adapters are a waste of money.
There is one partial caveat. Older supplies (from before the Pentium 4 era, especially) were typically built on the assumption that most power draw would be on the 3.3 V and 5 V rails, and relatively little would be on the 12 V rail. In modern systems, the processor and power video card lean heavily on the 12 V rail, and consequently newer systems need a stronger 12 V rail. Old power supply designs may have difficulty coping with this situation.
This can be somewhat touchy. In most cases, it will also Just Work. Plug the main connector in, leaving the extra 4 pins on the power supply's connector hanging free. This will not introduce any instability issues or other problems of that nature.
However, some motherboards have components very close to the main connector socket. These are generally capacitors used for power filtering. In some cases, they may block the area that the extra 4 pins would occupy. In this case, it is not a bad idea to buy an adapter; they cost about $10. The alternative would be to swap the power supply for one with the 20+4 pin connector (or one with the plain 20-pin connector, if need be), or to replace the motherboard.