What does non-reflex mean? And what exactly is a DSLR camera?

I've been using a canon powershot pro 1 for about 5 years now if not more but it has finally died (well not really - it has "mood swings". Most pictures come out out highly overexposed and when they're not over exposed or even just plain white, they come out all stripey. I've asked around but most camera places give me the classic "it would be cheaper just to buy a new one that get it repaired" story… -.-)
It was given to me by my dad when I was younger when he got a more professional level nikon and to be honest i never properly learn how to use it. I would like something a little more "tweakable" than your average point and shoot but still quite portable as i travel a lot everyday and aftr colleg i plan to travel even more! By tweakable i mean that even in manual mode on point and shoot cameras, you're still quite restricted and I want to learn how to use a camera properly like howthe ISO woks and the shutter speed and aperture etc.

I'm still saving up and still quite lost with what camera to pic. I'm looking at the Olympus PEN range but I really dislike that it has no viewfinder or built-in flash! In a review (http://www.dpreview.com/reviews/olympusep1/) it talks about a "non-reflex interchangeable lens camera [built] around a large sensor".so now I'm kind of confused as to what a single lens reflex camera is and what a non-relex camera is. Is there a such thing as a multi lens reflex camera? What makes an SLR an SLR?
Sorry for such a long-winded question, I can understand if answering it all would be too long, if so do you know of any good articles or websites I can read to learn about this all?

Thanks so much for your time!

P.S. I'm not a professional, I probably don't even qualify as vaguely amateur haha! I'm a student and I love photography and I want to learn, that's all. Thank you for your help.

Added (1). And what is "four-thirds" is this different to SLR?

Thank you, so are the mirrors just for seeing your image through the viewfinder or do they take any part in taking the final picture at all?

DSLR. Digital Single Lens Reflex. SLR cameras have a mirror that reflects the light through the lens into their finder. When the shutter button is pushed, the reflex action, quickly moves the mirror while the mirror out of the way while the shutter opens to allow light onto the film/sensor.

A Single Lens Reflex camera, digital or not has only a single lens assembly (containing several elements). Because of this it's possible by changing the light path to see in the viewfinder, exactly the image that will be captured on the film or sensor. So when you see your shot through the viewfinder the mirror is down and nothing is hitting the sensor. When you press the shutter release, the "reflex" kicks up the mirror out of the way, the viewfinder goes black, and the shutter fires exposing the film or sensor to the light. Finally the shutter closes, the mirror drops back down and you can see through the viewfinder again. So the mirror is there for you the photographer, but once you come to take your photo it has to move out of the way, taking no part in the actual capture. That is unless you use a Digital Single Lens Translucent Mirror camera (DSLT), but that's another story (see the SLT-A77 on Sony's site for details).

As for 4/3's it's a sensor size and lens lens mount specification. The largest sensor size you'll find on a DLSR is Full frame or FX, basically the sensor is the same size as that of a single frame of 35mm film. Next up is DX or APS-C, which is based on the frame size of negatives generated by the Advanced Photo System "classic". Slightly smaller again is the 4/3's sensor, the smaller size of this sensor allows for a smaller body and lens mountings for more compact cameras. Taken a step further a smaller version of the 4/3's mount the micro 4/3's allows even smaller cameras but still using the same 4/3's sensor. 4/3's cameras can be DSLRs but they don't have to be, and Sony has it's NEX range that uses the larger APS-C sensors in a compact body.