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Right now I'm just using an unmodified digital SLR when I take photos, but I've heard that ordinary DSLRs have a filter on them that reduces what it picks up on the low (red) end. How much of a difference does it make and is it worth messing with if I get really serious with my photography?

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Although the CCDs in DSLRs are quite sensitive to the deep red colors (including H-alpha), the camera manufacturers added a filter over that part of the spectrum to make the photos they take appear more like the images we see with our eyes (that aren't so sensitive to those wavelengths).

However, in astrophotography, H-alpha is important. Often, big gas clouds and nebulas are composed largely of hydrogen. When they are lit up by radiation from other stars, they re-emit a large portion of that energy as H-alpha light, because the lowest visible light band for hydrogen is H-alpha. The filter over the CCDs don't target H-alpha light of course, they just reduce the sensitivity of the CCD to that color--at the H-alpha wavelength, only 20-25 percent of the light actually gets through to the CCD.

Some DSLRs, such as the Canon 20Da were designed specifically for astrophotographers, where the filter was changed to let up to 70 percent of this light through. That camera has been discontinued however.

The filters can be modified but this can be tricky and expensive (especially if you brick your sensor). You are still able to photograph these nebulas, just not with the intensity that you would be able to without that filter.

Source: http://www.astropix.com/HTML/I_ASTROP/DSLR_HA.HTM

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