A reflex camera is simply one that has a mirror in its guts. It allows the light coming in from a lens to be reflected up into a viewfinder or onto ground glass so the photographer can see the image and adjust composition before taking it.
This is in contrast to, say, a view camera (the kind with bellows), where the light just goes straight through to the image plane/ground glass [so it appears upside down and reversed left-to-right]; a rangefinder, where you view the scene straight through a separate lens next to the one used to take the image; or a digital mirrorless or compact camera, where light goes directly to the sensor and sensor data is conveyed to the viewfinder/LCD.
Most reflex cameras are in one of two categories: SLR or TLR.
Single-lens reflex (SLR) cameras are by far the most prevalent and have supplanted the older TLR cameras and these days come with digital sensors. The lens reflected by the mirror up into the viewfinder is then passed through a pentamirror or pentaprism to flip the image around so it appears in the eye-level viewfinder the same way the scene does, rather than upside down and left-to-right reversed.
Twin-lens reflex (TLR) cameras use two lenses: one for taking an image, the other for the viewfinder. The viewfinder lens is usually stacked over the taking lens. So you have two separate lightpaths and parallax, like with a rangefinder. But the viewfinder lens's light is projected up onto ground glass, and you compose looking down at the glass at waist level (i.e., a waist-level viewfinder). But, the image on the glass is reversed left-to-right. TLRs never really went digital, and mostly remain a specialist tool for medium-format film enthusiasts.