DSLRs are the most common type of camera used by professional photographers and by consumers who want better quality and more flexibility than what their camera phone can provide.
DSLR stands for Digital Single Lens Reflex. "Single Lens Reflex" refers to the structural design that is specific to this type of camera. "Digital" refers to the fact that this camera records data on digital media instead of onto a roll of film.
SLR cameras are structurally very similar to DSLRs and can even use the same lenses in most cases, but they use 35mm film instead of SD or CF cards.
My dusty well-loved DSLR in all its glory. Nikon D750. Attached is a 50mm 1.4 lens.
DSLRs and SLRs have the ability to be used with a variety of lenses which is one reason why they are so popular. Nikon, Canon and other major manufacturers have built a wide variety of lenses at different price points and focal lengths to be easily used with DSLR and SLR camera bodies. Almost anything you'd like to photograph can be done so with a DSLR (or SLR) and the appropriate lens.
If you want to buy a DSLR, you can get your hands on a new one from as little as $400 or for as much as around $6,000. Professional grade camera bodies start at around $1600. Regardless of price point, they all basically work the same way. The upgrades found at the higher price points are improvements you will only notice and appreciate if you shoot A LOT. So lower price points are just fine for beginners or casual photographers. The other main difference I notice is that the lower price models are built to anticipate the needs of amateurs, not professionals, and the way they design the buttons and menu reflect that to some extent. The placement of the controls on entry level DSLRs is often a tad less conducive to shooting manually. This in no way will affect the quality of your images so its really no big deal - and when you learn to shoot manually it will feel like second nature to you no matter which model you have.
My favorite places to buy photography equipment are listed below, with Bhphotovideo being my favorite. BH Photo Video has awesome customer care and they also have a used department where you can purchase reliable used and refurbished equipment at a good discount which I LOVE.