Like your smartphone eurorack modules want to have a certain voltage. In your smartphone this is 5V DC thanks to EU regulation. In the world of modular synthesizers the most widespread standard is the Eurorack standard: ±12V. This means the usual Eurorack module wants to see three voltages on their powerconnector: +12V, -12V and Ground (0V).
The positive and negative 12V voltages are called "Rails".
There is another value you will find on the back of your smartphone charger: current. This current is usually measured in Ampere (A) or Miliampere (mA). If your powersupply "has" 1A, this means you can connect anything to it that draws less current than 1A. If you connect something that draws more current than your powersupply can deliver, you can damage either your module, or the powersupply. If your powersupply is good, it will just lower the voltage in order to meet the current draw needs of whatever is connected to it.
A power supply (short: PSU) is a circuit that takes a input voltage (for example 230V AC from the mains socket in the wall) and converts this into a voltage suitable for our needs. The typical small phone charger for example converts 230V AC to 5V DC (represented by this symbol ⎓) and has two outputs: +5V⎓ and Ground (0V⎓). As mentioned, in the Eurorack world we need a symmetric supply voltage of ±12V⎓. This means we need a powersupply which has three outputs (+12V⎓, Ground and -12V⎓).
Aside from the voltage each PSU has a maximum current it can deliver to whichever circuit is connected. As long as the stuff we hook up to the powersupply stays within the limits of the powersupply we are safe. A typical Eurorack module may have an average current draw of about 70mA, but usually manufacturers will tell you how much each module draws on each rail (e.g. 70mA on +12V and 50mA on -12V). You can also create an example rig on Modulargrid.net and read the total current needed on each rail. If you construct your own modules and you need to know how much current it draws, you can use a multimeter (switched to mA) in Series (!) with the rail you want to measure in order to measure the current draw. Or if you have access to a Lab power supply, these usually display how much current is drawn from them at a given moment.
The PSU will be connected to a Eurorack Power Distribution Board, which has connectors for the module power cables (see power accessories). Connecting the module to the Power Distribution Board can be potentially dangerous for some modules (and in some cases for the powersupply), if the polarity is flipped (=you plugged in the cable the wrong way). Make sure to know 100% which voltage you are plugging into the module and definitely consider to measure it with a voltmeter before you plug it in.
Linear Power Supplies
It is generally desirable for a powersupply (especially in audio circuits) to be linear. This means the powersupply should deliver the same voltage on both rails no matter what the load (momentary current draw of our modules) and no matter which voltage the power grid currently has (in Europe's 230V AC grid there is a typical deviation of ±25V AC, which means the voltages coming from the powerplug can vary between 205V AC and 255V AC). An easy and commonly used way to deal with these unstabilities is the usage of Voltage Regulator chips (e.g. LM317 or LM350 for positive and LM337 for negative Voltages) for each rail. These chips will try their best to keep the output voltage rock solid where it is, as long the current drawn is within its limits. The only downside of Voltage Regulators is: they need a higher voltage on the input than on the output. So if you want to get a stabilized ±12V⎓ from a pair of LM317 and LM337 you need at least ±13.7V⎓ input Voltage (because a voltage regulator can only reduce voltage and it needs a certain amount of voltage to regulate).
Most bigger powersupplies utilize a toroid transformer ("Ringtransformator" in German), which (usually) has 2 input cables for 230V AC and 4 output cables which can be used to get ±12VAC and a centered Ground (0V) out of them. After the ±12V there is usually a bridge rectifier to convert it to DC and a pair of Voltage Regulators with a bunch of capacitors to stabilize the Voltage and filter potential noise which might come in through the powerline. The drawback of this is the size and weight of the transformer, which must also be placed inside the Synthesizer's case or in another enclosure because there is dangerous 230V AC on one side of the transformer and bad wiring can potentially kill someone here. For a schematic and build look at the simple PSU described here.
Cheap DIY Hacks
The sometimes falsely proposed "cheap hack" of using a voltage divider (made out of two resistors or two diodes) to get ±12V⎓ with a virtual ground from a 24V⎓ source does indeed produce ±12V⎓ if measured. But the virtual groundpoint is inherently unstable which means with the slightest asymmetrical load (current draw) it will shift the 0V of your supply, you might then no longer have ±12V⎓ and 0V⎓ but something like +18V⎓ 0V⎓ -8V⎓, which can potentially destroy your modules, but is in any case audible in the output sound. One way to stabilize a voltage divider is proposed here.
Buy something commercially available
Because of the weight, the danger and liabilities connected to implement a transformer yourself, a good solution might be to use a small DC powersupply (similar to a notebook powersupply) produced by professionals and to additionally stabilize the voltages. Or you might just buy a complete Low Voltage solution like the μZeus by Tiptop Audio, the Row Power by 4ms, the Silta by Frap Tools, the Deluxe Power Module by Synthrotec or various others.
Another way to generate a negative Voltage is to use a special IC like a LTM8049, which can boost lower voltages with setable switching frequencies and outputs a setable frequency. [Find cheaper IC here?]
Connecting a PSU
Electrically non-skilled persons are advised to leave the dangerous part of this work to a professional.
If this professional however needs to know how to do the wiring, an example wiring diagram is shown in this doepfer wiring pdf.