I think I found it on amazon.
Well, to fully understand this requires a bit of study of the actual functions of your equipment. And to be candid, having a basic grasp of the theory behind what your gear is doing is knowledge that will be very helpful as you progress!
[And a head's up: a lot of this information is covered in the Yamaha Sound Reinforcement Handbook, 2nd Edition. If you don't have this book, you should buy it, if for no other reason than it makes girls think you're smarter.]
So, as /u/jasmith-tech stated, "a mixer is a mixer." And to get it super simple, a mixer takes a certain input voltage, increases it to a level that you can use it, and then outputs it along the signal chain. Notice I said voltage and not volume. [Common misconception that amplifiers produce watts. Nope. They produce voltage that then, when applied against the resistance (ohms) of the speaker, result in watts being measurable.]
There's two terms you'll hear- microphone level, and line level. On a basic mixer, you'll commonly see microphone-level inputs available as XLR ports, since they're typically intended for you to plug a microphone into, and then 1/4" inputs, designed to receive line-level signals, such as those you'd find on keyboard outs, console outputs, and lots of other signal processing units. "Mic Level" extends from no signal up to about -20dBu (77.5 mV). "Line Level" extends from -20dBu to +30dBu (24.5V).
So, boiling it down, and you'll probably catch on now- when you're plugging your iDevice directly into the microphone input on the powered speaker, the input is not being driven with sufficient voltage to then get enough increase in order to send enough voltage into the built-in amplifier! But, by connecting your iDevice into your mixer, you're gathering that voltage increase and building it to a level that's useful for the speaker.
This is also the reason why you don't get a useful volume if you connect your passive-speaker-with-1/4"-inputs to your passive-mixer-with-1/4"-outputs. Again, the speaker isn't being given enough voltage to do anything useful with. It'll WORK, just super low level.
When you mentioned powered versus passive mixers, the only difference between them is that a powered mixer has that voltage amplifier built in (with speaker-level outputs designed to connect to your speaker), whereas the passive mixer relies on an external amplifier to accomplish the same task. So, a powered mixer connects directly to a passive speaker, whereas a passive mixer can connect either to an external amplifier (typically via XLR or 1/4" cable to the amplifier), and then to passive speakers (typically via SpeakOn cabling, but you can find heavy duty 1/4" cable doing the same thing) OR to a powered speaker (again, typically via XLR cable, with the amplifier built into the speaker cabinet). Which method is best for your circumstance is entirely up to you. :-)
Hope this helps! And yeah, go grab that book if you'd like to get a really solid foundation to the underlying physics of the audio industry. There's a reason it's one of the "bibles" that is commonly seen on people's shelves.
I'm sure that's OK- what with the spelling and grammatical errors in the survey, English is probably also not the OP's first language.
Jess, I'd start with The Sound Reinforcement Handbook to get some good research material to start with. By actually doing the research, you'll learn a LOT more than just asking people to "fill out surveys."
The Yamaha Sound Reinforcement Handbook
This is basically the soundman's bible. Sold for an aspiring live sound engineer, has much more than just advice for live sound guys - covering everything from microphones to acoustics to basic electronics to handy rules of thumb to MIDI, all written to be relentlessly pragmatic. It even has a handy appendix covering logarithms.
This isn't the book to give you the final "20%" of knowledge on anything it covers - but it will help you on your way to the first 80% a lot more quickly than most other writing on anything related to semi/pro audio, and pretty much every expert in the field is at least familiar with it, if they don't own a copy.
There seem to be plenty of people interested in music and audio around here, so hopefully someone finds the unusual reference useful.