The current-producing capacity of the power supply must be more than the worst-case current rating of the pedal(s). If you have a 60mA power supply, and pedal rated at 25mA will work fine, and will draw 25 mA. An 80mA pedal might work, but when it draws its peak current, weird things could happen. A 100 mA pedal probably won't work at all.
When overloaded, some power supplies will "sag", putting out less voltage as the load increases. Others have a "crowbar" protection circuit, which senses the overload and shuts off the power output completely until the over load is cleared.
Purely analog effects, like fuzz, distortion, OD, boosters will run okay on low voltage or saggy power supplies and might sound different. Effects that use a clock circuit (modulation) probably won't work on insufficient power supplies. Digital effects definitely won't work.
Pay attention to whether the power supply puts out AC or DC as well. There are AC wall warts out there, design for products that put the rectification and filtering in the device itself, rather than the wall wart. In general, pedals need 9 volts of filtered, regulated power.
