Batteries, wall wart power supplies and laboratory power supplies are all means to power your handy-work. There are several specifications you need to take into account when working with external power supplies, which can help you to keep your components in one piece when experimenting.
Voltage: fixed vs adjustable
For powering an Arduino, a fixed voltage adapter can be used. For powering your prototype, it can come in handy to have an adjustable output voltage. Modern electronic circuits often work on a 3.3 Volt or 5 Volt DC supply voltage (sometimes even as low as 3 Volt). That is why USB plugs have a 5 Volt DC power connection as well.
Current (and current limit)
When you are experimenting or are not experienced in electronics yet, you can set a current limit on a laboratory power supply. In that way you know that your circuit can’t draw more current than the limit you set, protecting your components in case of a short circuit or error.
Batteries have a capacity and current rating, as well as a cell voltage. For Alkaline cells, 1.5 Volts indicates a full cell. Where rechargeable (NiMH and NiCd) batteries only reach 1.2 Volts when full. Lithium-ion batteries have a typical cell voltage of 3.7 Volt. When a battery depletes, the cell voltage drops accordingly. Batteries deplete according to the current that is drawn: low current -> long lifetime. High current -> short lifetime. That is why battery capacity is specified in Ampere hour (Ah). Equal to the amount of continuous current multiplied by the discharge time, that a battery can supply energy, before it is exhausted. In case you need a 5 Volt supply, consider the use of a USB power bank. These are available in many different capacities. Charging and discharging is controlled via an internal circuit, improving safety. Modern powerbanks may shutdown if not enough current is used. Adding an extra resistor between 5 Volt and GND connection may help in that situation to prevent the powerbank from shutting down.
Wall wart power supplies (unregulated vs regulated)
Caution should be observed when choosing a wall wart power supply. Unregulated types can output voltages a lot higher than their specified voltage, possibly destroying any connected electronics that is not rated for that voltage. When a low current is drawn from the supply, the output voltage is usually higher than when it is under load. Measuring with a multimeter and a dummy load (resistor or lamp) can give you an idea of the behavior of a power supply. The current rating on the power supply should always be observed, and never overloaded. Overheating and short circuits could be the result.
Regulated DC power supplies give off a constant voltage (within their specified current range), regardless of the current that is drawn from the power supply.
Laboratory DC power supplies
Laboratory power supplies are regulated by standard. Most laboratory power supplies have an adjustable output voltage and current limit setting that help you experiment safely, without blowing up parts.