After getting a customer complaint (the only one I had in two and a half years), I was promptly "demoted." The upper management kept referring to it as a "demotion" and it kind of was since they were forbidding me from doing a lot of things that I would normally do as a supervisor, basically knocking me down to a regular employee. However, my title and pay did not change as a result of this. My name tag still said "supervisor" and I was still given keys. What do I take from this / what does it mean? Do they not have a real reason to demote me and just don't like me?
I don't want to put you on the spot, but it would be very difficult to say, without knowing more details, such as the nature of the mistake, its perceived severity, the result and impact of the mistake on the customer and company, etc.
Since I don't want you to bear your soul to everyone, I would advise you to speak to your supervisor and ask what the plan is, how this will affect you, and most importantly, how you can correct/avoid the mistake and regain the trust. it may turn out that this is part of documenting the issue, which is important to affecting any changes that they feel might need to take place. Ultimately, you just need to bring it up, be receptive and non-defensive regarding whatever feedback might come your way, with the goal of finding out what you can do about it.
Whatever you do, don't sit and wonder. Take some action, and if you feel you're in the dark, ask the right questions to get some light shed on the situation. I don't want to try to explain the behavior of your supervisor, because there isn't enough information, and everyone's approach is different when working with team members that make mistakes.
First be grateful they didn't reduce your pay.
What to take from it is that you messed up but not so badly that you can't be put back into your previous level of trust. The key is to take responsibility for whatever happened and look for ways to change to ensure it doesn't happen again. The worst thing to do is to think, "it wasn't my fault and they are out to get me" and theerfore to whine about how bad things are. Your attitude speakes volumes here.
It is possible to get demoted and be repromoted in the same organization. It just happened where I work in fact. It is also possible to move on and get promoted elsewhere. Your career isn't ruined and you have the chance to grow professionally from this.
normalocity is right that you should just talk to your supervisor. However, I can give a few guesses as to their motivation:
1) They wanted to make it clear to you that you did something wrong, but didn't think the offense warranted a full demotion with loss of title and pay.
2) They're hoping you'll improve your behavior and plan on giving you the old responsibilities back once that happens, or giving you a real demotion if it doesn't happen - for now they're just waiting to see and saving themselves the paperwork.
3) They didn't think whatever you did was very bad at all, but needed to be able to say they did something in response to the complaint when asked (by the higher management, or the customer in question), so they picked an option that's least harmful to you.
In either of the first two cases, it's important that you show effort to improve if you want to avoid more severe consequences; in the last case, it's still important that you at least that you not get any more complaints (though I realize that's not entirely under your control).