What makes dslr cameras better than compact digital cameras?

I just ordered the nikon d3100 which i believe has about 14.1 mega pixels. My mom just got an olympus compact digital camera that has 14 megapixels, but the quality is horrible. I know that the d3100 is supposed to have great quality pictures, but they have basically the same amount of megapixels, so what makes it a better quality than the olympus?

p.s. For all i know megapixels mean nothing at all… I don't know much about cameras lol i'm just curious what makes them so different.

Megapixels really don't effect the quality when using a DSLR.
My Dslr is 15.1 megapixels, but i only use 8 because i put it on medium sized pictures. And when i shoot in Raw i notice no difference.
DSLR's are wonderful, they really let you completely control your camera in every aspect, shutter speed, ISO, exposure, depth of field, size, aperture, etc. Plus the sensors in a DSLR are much bigger, smaller sensors significantly reduce the quality of an image. I'd much rather have a camera with less megapixels and a larger image sensor than the other way around.

Principally because of the larger sensor - squashing lots of MP onto a small sensor (what happens in a compact) leads to poor image quality, especially in low light. The same number of pixels on a larger sensor (in a DSLR) will give better quality images.

other advantages of a DSLR are;
full manual control of exposure,
much shorter shutter lag,
interchangeable lenses, (which mean they function better than a 'jack of all trades' lens),
better optical quality lenses,
ability to shoot in raw format,
presence of a hotshoe (which allows control of off camera flash)

The SLR has a physically larger sensor.So, its 14 megapixels don't have to work as hard, to use a metaphor. It's individual pixel sites are bigger. This equates to better light sensitivity and dynamic range. The SLR probably has a better image processor too. This is important, but not generally recognized because it's an abstract concept. Furthermore, your internal data format for files is probably 12 or 14 bits per color channel. The point-and-shoot might be the jpeg standard of 8 bits. The optics in the lenses for SLRs are also better as a rule.

Your DSLR is a larger format camera than your Mom's compact. Your sensors have the same number of pixels, but the compact's sensor is tiny compared to yours. In this example you can see the Nikon DX format sensor compared to the compact's 1/2.5 sensor. Because of the size difference, the compact's image must be increased many more times than the DX sensor just to get an image large enough to view, and about 15X just to get to the size of the DX sensor. Every time you increase an image size you also magnify any flaws in the image. Camera format also plays a role in other capabilities of the camera.
There are many factors that affect image quality between cameras; construction, lens quality, sensor design, processor and software design.

Megapixels have nothing to do with quality; they are just a measure of the resolution the camera takes pictures at. So basically, more megapixels just means you can crop more and/or print larger.

The main reasons DSLRs offer superior image quality to compact digicams are as follows:

-Larger sensors
-Higher quality sensors (most digicams use CCD sensors, while most DSLRs use CMOS)
-Better lowlight performance at high ISO settings (a result of the above two things)
-More flexibility offered with lens choices
-Almost any DSLR lens is sharper than almost any point & shoot lens
-More control over the picture (RAW, manual exposure controls, etc.)
-More control over depth-of-field (point & shoots have such short physical focal-lengths that it's tough to get a shallow depth of field using one)
-Faster focusing, even with slow-focusing lenses

The thing is, tons of people buy a DSLR and flip it to Auto mode and go on their merry way, then are disappointed when they don't immediately have professional results. I realize that people see professionals carrying DSLRs and then see their images, and assume the difference is all in their camera. But the difference has very little to do with the equipment. Professional-quality images can really be generated by any camera. I've seen it done on an iPhone. What makes up a professional photo is use of composition, lighting, the correct settings, good exposure, just enough optimizing in post-processing, etc.

The DSLR's sensor is larger. Same number of pixels, but the pixels are larger, which means they can gather light more efficiently.

Versatility, resolution and quicker response time.