Forgive me, as I am far from an electrical engineer, but when rectifiers were explained to me I found that a rectifier allows current only in one direction, which allows it to filter out negative voltage of an AC power source and only output the positive, but where does this negative voltage go? Is it just lost in the conversion process? If so is there a circuit that can "recycle" this "lost" negative voltage?
The simple case you describe is a half-wave rectifier circuit, configured for a positive DC voltage output. In the most basic sense, the rectifier diode acts as a switch, either allowing electrons to flow through it (on) or not (off). During the negative half-cycles of the AC input, there is no electron flow. It is the same as you flipping a switch on and off 60 times each second, perfectly timed for the positive half cycles.
Current (electron motion) happens only when caused by external conditions. A battery, or a wall outlet, sits there with a potential difference (voltage) between its terminals. If there is no connection between the terminals (an electric load like a light, motor or whatever), no electrons flow; there is no current.
Pedantic note: "current flow" is not correct; current is the motion (flow) of electrons. Blah blah x 10^18 electrons per second moving past a point is called 1 ampere.