DSLRs are a subset of SLRs.

In other words all DSLRs are also SLRs, but not all SLRs are DSLRs.

SLR is an abbreviation for Single Lens Reflex. What this means is that composition of the scene, focusing, and actually recording the image are all done trough the same lens. The word reflex relates to the way a mirror is used to view the scene and focus it through the viewfinder or a focusing arrray and the way it is flipped up out of the way fractions of a second before the image is recorded. An SLR may use either film or a digital sensor as the recording medium.

DSLR means Digital Single Lens Reflex and inlcudes only SLRs that record the image with a digital sensor.

There are also some mirrorless and translucent mirrored interchangeable lens digital cameras that can be used much like a DSLR. The translucent mirrors do not move. Instead they only reflect a fraction of the light to the viewfinder and the rest is allowed to pass through to the lens. Most mirrorless interchangeable lens cameras use the main image sensor for composing the scene and focusing as well as recording the image. Many still have some form of mechanical shutters, many others do not.

Digital Sensors and Film are two distinct recording mediums which both display unique characteristics when recording an image. The way an image is processed, or edited for viewing, is also much different. Both have characteristics that can sometimes be an advantage and sometimes be a disadvantage depending on the subject matter and the qualities the photographer wishes the final image to display. In the end it is a matter of personal taste. For more on film vs. digital please see this question, this question, and this one.